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ELEMENTS OF ALGEBRA: 



ON 



THE BASIS OF M. BOURDON; 



EMBRACING 



STURM'S AND HORNER'S THEOREMS, 



PRACTICAL EXAMPLES. 



BY CHARLES DAYIES, LL.D. 

AVraOR OF ARITHMETIC, ELEMENTARY ALGEBRA, ELE:irENTART GEOMETRT, PRAOTTOM 

aKOMETRT, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE AND 

ANALYTICAL GEOMETRY, ELEMENTS OF DIFFERENTIAL 

AND INTEGRAL CALCULUS, AND A TREATISE 

ON SHADES, SHADOWS AND PER- 

• SPECTIVE. 



NEW YORK: 
A. S. BARNES & BURR, 51 & 53 JOHN STREET 

SOLD BY B00KSELLBK3, GENERALLY, THEOUGHOUT THE UNITED STATES. 

1864\ 






iiabies' primaro ^rit|)mctic and <!i:ahle=33oofe— Designed for Beginners; 
containing the elementary tables of Addition, Subtraction, Multiplication, 
Division, and Denominate Numbers ; with a large number of easy and prac- 
tical questions, both mental and written. 

IDsbies' jFtrst 3Les£ons in ^rittmctic— Combining the Oral Method with the 
Method of Teaching the Combinations of Figures by Sight. 

IBabics' J-ntcUectual ^vitl)m£?tc— An Analysis of the Science of Numbers, with 
especial reference to Ment?J Training and Development. 

23abies' ISSc'm ,Scf)Ool ^rttDmetic — Analytical and Practical. 

B.e» to Babies* Keto .Sctiool gtrit|)mctic. 

Sables' eSrammac of ^ritljmetic — An Analysis of the Language of Nmnbers 
and the Science of Figures. 

iSabies' "Nt'm 2Inibecsiti) ^ritt)inctic — Embracing the Science of Numbers, and 
their Applications according to the most Improved Methods of Analysis and 
CancelLition. 

Kej to iBabics* Ncto sanibetsi'ti) ^ritljmettc. 

33abirs' Hlcmcntauy gllgcftra— Embracmg the First Principles of the Science. 

I^?r) to Babies' SlancutavD Algebra. 

Babies' iSIementarj ©feometr^ and STrfflononietrg — With Applications in 
]*.Iensuration. 

Babies' 3,9i"^ctieal I^atjematfes—With Drawing and Mensuration applied to 

the Mechanic Arts. 

Babies* gEufbersitij ^Iflcfira— Embracing a Logical Development of the 

Science, with graded examples. 

Babfes' 3Souvtioit's .^Iflcbra— Including Sturm's and Horner's Theorems, 

and practical examples. 

Babies' S,rf entire's CKcometr^ auU ^rfsonometrj— Revised and adapted to 
the course of Mathematical Instruction in the United States. 

Sabfrs' lEUmivAs of Suiberfnij and "Xabfuation — Containing descriptions 

of the Instruments and necessary Tables. 

Babfes* Slmlptfeal ^eomctvn— Embracing the Equations of the Point, the 
Straight Line, the Conic Sections, and Surfaces of the first and second order. 

Babies' Bifferential and J-ntegral €:aleulus. 

Babies* Bescvmtibe CS^eometro— With its application to Spherical Trigonomt- 
try, Spherical Projections, and Warped Surfaces. 

Babies' <Si)ai3es, S|)nLio\DS, and ^^^erspectibe. 

Babie.s' 2.02fc anti mVdtv of |Hat1)ematies— With the best ntctliods of In- 
struction Explained and Illustrated. 

Babies' anU ^aeek's fanttjematieal Bfetfonarp antr CTajcIopclifa of |iJatl)e* 
mat cal Science — Comprising Definitions of all the terms employed in 
Matheinatics — an Analysis of each Branch, and of the whole, as forming a 
single Science. 



Entered according to Act of Congress, in tlie year one thousand eight hundred and fifty- 
one, by Chapxes Davies, in the Clerlt's Ofiice of the District Court of the United States 
for the Southern District of New York. 



"William Denyse, Stebeotypeb and Electrotyper, 1 S3 William Street, New York. 



PREFACE 



The Treatise on Algebra, by M. Bourdon, is a work 
of singular excellence and merit. In France, it liaa 
long been one of the standard Text books. Shortly after 
its first publication, it passed through several editions, 
and has formed the basis of every subsequent work on 
the subject of Algebra, both in Europe and in this country. 

The original work is, however, a full and complete 
treatise on the subject of Algebra, the later editions 
containing about eight hundred pages octavo. The time 
which is given to the study of AlgebraJ in this country, 
even in those seminaries where the course of mathe- 
matics is the fullest, is too short to accomplish so volu- 
minous a work, and hence it has been found necessary 
either to modify it essentially, or to abandon it alto- 
gether. 

In the following work, the original Treatise of Bourdon 
has been regarded only as a model. The order of ar- 
rangement, in many parts, has been changed; new rules 
and new methods have been introduced : the modifica- 
tions indicated by its use, for twenty years, as a text book 



4 PREFACE. 

in the Military Academy have been freely made, for 
the purpose of giving to the work a more practical 
diaracter, and bringing it into closer harmony with the 
trains of thought and improved systems of instruction 
which prevail in that institution. 

But the work, in its present form, is greatly indebted 
to the labors of William G. Peck, A. M., U. B. Topo- 
graphical Engineers, and Assistant Professor of Mathe- 
matics in the Military Academy. 

Many of the new definitions, new rules and improved 
methods of illustration, are his. His experience as a 
teacher of mathematics has enabled him to bestow "upon 
the work much valuable labor which will be found to 
bear the marks of profound study and the freshness of 
daily instruction. 



FiBHKILL liANDDie, 

May, 1858. 



CONTENTS. 



CHAPTER I. 

^ DEFINITIONS AND PRELIMINARY REMARKS. 

ABTIOLIA 

Algbsra — Definitions — Explanation of the Algebraic Signs 1 — 28 

Similar Terms — Reduction of Similar Terms ,. 28 — 80 

Theorems — Problems — Definition of — Problem 30 — 31 

CHAPTER 11. 

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. 

Addition — Rule 31—36 

Subtraction — Rule — Remark 36 — 41 

Multiplication — Rule for Monomials and Signs 41 — i& 

Rule for Polynomials - 45—45 

Remarks — Theorems Proved 46 — 49 

Division of Monomials^— Rule 49 — 63 

Signification of the Symbol af^. 53 — 55 

Division of Polynomials — Rule 55 — 58 

Remarks on the division of Polynomials 58 — 59 

Of Factoring Pylynomials 59 — 60 

When m is entire, a"* — 6™ is divisible by a — b 60 — 62 

CHAPTER III. 

ALGEBRAIC FRACTIONS. 

Definition — Entire Quantity — Mixed Quantity 62 — 68 

Reduction of Fractions. : : , 68 — 69 

To Reduce a Fraction to its Simplest Form.... 68 — I. 

To Reduce a Mixed Quantity to a Fraction 68 — IT. 

To Reduce a Fraction to an entire or Mixed Quantity. 68 — III. 

To Reduce Fractions to a Common Denominator 68 — IV 

To Add Fractions 68— V. 

To Subtract Fractions , , 68— VI 



6 CONTENTS. 

ARTICLIS 

To Multiply Fractiors 68— VIL 

To Divide Fractions 68— VIH 

Results from adding to both Terms of a Fraction 70 — 71 

Symbols 0, » and J 71— *^2 

CHAPTER IV. 

EQUATIONS OF THE FIRST DEGREE. 

Definition of an Equation — Different Kinds — Properties of Equations 72 — 77 

Solution of Equations - 77 — 78 

Transformation of Equations — First and Second 78 — 80 

Resolution of Equations of the First Degree — Rule 81 

Problems involving Equations of the First Degree 81 

Equations with two or more Unknown Quantities 82 — 83 

Elimination — Bj Addition — By Subtraction — By Comparison 83—88 

Problems giving rise to Simultaneous Equations...... Page 96 

Indeterminate Equations and Indeterminate Problems 88 — 89 

Interpretation of Negative Results 89 — 91 

Discussion of Problems 91 — 92 

Inequalities 92—93 

CHAPTER V. 

EXTRACTION OF THE SQUARE ROOT OF NUMBERS, OF ALGEBRAIC QUAN- 
TITIES. TRANSFORMATION OF RADICALS OF THE SECOND DEGREE. 

Extraction of the Square Root of Numbers 93 — 96 

Extraction of the Square Root of Fractions 96 — 100 

Extraction of the Square Root of Algebraic Quantities 100 — 104 

Of Monomials ' 100—101 

Of Polynomials 101—104 

Radicals of the Second Degree 104 — 106 

Addition and Subtraction— Of Radicals....* 106—107 

Multiplication, Division, and Transformation 107 — 110 

CHAPTER VI. 

, EQUATIONS OF THE SECOND DEGREE. 

Equations of the Second Degree 110 — 112 

Incomplete Equations — Solution of 112 — 114 

Solution of Complete Equations of the Second Degree 114 — 115 

Discussion of Equations of the Second Degree 1 15 — 117 

Of the Four Forms 1 17—121 

Problem of the Lights 121—122 

Of Trinomial Equations 122—125 

Extraction of the Square Root of the Binomial a dlz^/ b 125 — 126 

Equations with two or more Unknown Quantities.... ..126 — 128 



CONTENTS. 7 

CHAPTER VII. 

FORMATION OF POWERS, BINOMIAL THEOREM, EXTR A-CTION OF ROOTS 
OF ANY DEGREE WHATEVER. OF RADICALS. 

Formation of Powers, 1 28 — 13C 

Theory of Permutations and Combinations, ,130 — 136 

Binomial Theorem 13G— 141 

Extraction of the Cube Roots of Numbers. 141 — 142 

To Extract the 7i'* Root of a Whole N'umber 142—144 

Exti-action of Roots by Approximation 144 — 145 

Extraction of the n'-^ root of Fractioias 145 — 146 

Cube Root of Decimal Fractions 146 — 147 

Extraction of Roots of Algebraic Quantities 141 — 148 

Of Polynomials 148—150 

Transformation of Radicals 150 — 15^ 

Addition and Subtraction of Radical^ . 155 — 155 

Multiplication of Radicals , 156 — 151 

Division of Radicals. 157 — 158 

Fonuatiuu of Powers of Radicals , 158 — 159 

Extraction of Roots 159 — 160 

Different Roots of the Same Power 160—162 

Modifications of the P^ules for Radicals 162 — 164 

Theory of Fractional and Negative Exponents 164 — 171 

CHAPTER VIII. * 

OF SERIES. ARITHMETICAL PROGRESSION. GEOMETRICAL PROPORTION 

AND PROGRESSION. RECURRING SERIES. BINOMIAL FORMULA. 

SUMMATION OF SERIES. PILING OF SHOT AND SHELLS. 

Series Defined. . . : 171—173 

Arithmetical Progression — Defined 172 — 173 

Expression for the General Term 174 — 176 

Sum of any two Terms 1 75 — 176 

Sum of all the Terms 176—177 

Formulas and Examples 177 — 181 

Ratio and Geometrical Proportion 181 — 186 

Geometrical Progression — Defined » 186 — 187 

Expression for any Term .v». 187 — 188 

Sum of n Terms — Formulas and Examples 188 — 193 

Indeterminate Co-efficients. . , 193 — 199 

Recurring Series ^ ^ 199 — 202 

Genei-al Demonstration of Binomial Theorem 202 — 204 

Applications of the Binomial Formula 204 — 208 

Summation of Series . 208 — 209 

Method of Differences 209—210 

Pilmg of Balls 210—21* 



8 CONTENTS. 

CHAPTER IX. 

CONTINUED FRACTIONS. EXPONENTIAL QUANTITIES. LOGARITHMS.^ 

Continued Fractions 215 — 224 

Exponential Quantities , 224 — 227 

Theory of Logarithms. 227 — 229 

General Properties of Logarithms 229 — 236 

Logarithmic Series — Modulus 236 — 241 

Transformation of Series. 241 — 242 

Of Literpolation. 242—243 

Uf Interest 243—244 

CHAPTER X. 

GENERAL THEORY OF EQUATIONS. 

General Properties of Equations 244 — 251 

Composition of Equations 251 — 252 

Of the Greatest common Divisor 252 — 262 

Transformation of Equations 262 — 264 

Formation of Derived Polynomials 264 — 266 

Properties of Derived Polynomials 266 — 267- 

Equal Roots , 267—270 

jaimination. 270—275 

CHAPTER XI. 

BOLUTION OF NUMERICAL EQUATIONS. STURm's THEOREM. CARDAn's 

RULE. HORNEr's METHOD. 

General Principles 275 — 277 

First Principle 277 — 279 

Second Principle 279—280 

Third Principle 280—281 

limits of Real Roots 281 — 284 

Ordinary Limits of Positive Roots 284 — 286 

Smallest Limit in Entire Numbers 285 — 286 

Supeiior Limit of Negative Roots — Inferior Limit of Positive and 

Negative Roots 286 — 287 

Consequences 287 — 293 

Descai-tes' Rule 293 — 295 

Commensurable Roots of Numerical Equations 295 — 298 

Sturm's Theorem 298—308 

Cardan's Rule .«. 308 — 309 

Prelirainaries to Homer's Method 309 — 310 

Multiplication by Detached Co-efficients 310 — 311 

Division by Detached Co-efficients 311 — 312 

Synthetical Division 312 — 313 

Method of Transformation 313—314 

Homei B Method > 814 



INTRODUCTIOIf 



Quantity is a general term applicable to everything which 
can be increased or diminished, and m.easured. There are two 
kinds of quantity; 

1st. Abstract quantity, or quantity, the conception of which 
does not involve the idea of matter ; and, 

2dly, Concrete quantity, which embraces every thing that is 
material. 

Mathematics is the science of quantity ; that is, the science 
which treats of the measurement of quantities, and of their 
relations to each other. It is divided into two parts : 

1st. The Pure Mathematics, embracing the principles of the 
science and all explanations of the processes by which these 
principles are derived from the abstract quantities, Number 
and Space : and, 

2d. The Mixed Mathematics, embracing the applications of 
these principles to all investigations involving the laws of 
matter, to the discussion of all questions of a practical nature, 
and to the solution of all problems, whether they relate to 
abstract or concrete quantity.* 

*Davies' Logic and Utility of Mathematics. Book IL 



10 INTRODUCTION. 

There are three operations of the mind which are Imme 
diately concerned in the investigations of mathematical science ; 
1st). Apprehension; 2d. Judgment; 3d. Reasoning. 

1st. Apprehension is the notion, or conception of an idea 
in the niind, analogous to the perception hy the senses. 

2d. Judgment is the comparing together, in the mind, two 
of the ideas which are the objects of Apprehension, and pro 
uouncing that they agree or disagree with each other. Judg 
nient, therefore, is either affirmative or negative. 

3d. Reasoning is the act of proceeding from one judgment 
to another, or of deducing unknown truths from principles al- 
ready known. Language affords the signs by which these opeia- 
tions of the mind are expressed and communicated. An appre 
kension, expressed in language, is called a term; a judgment, 
expressed in language, is called a proposit'on ; and a pro^ s?s 
of reasoning, expressed in language, is called a demonsira 
tion* 

The reasoning processes, in Logic, are conducted usually by 
means of words, and in all complicated cases, can take place 
in no other way. The words employed are signs of ideas ^ 
and are also one of the principal instruments or helps of 
thought; and any imperfection in the instrument, or in the 
mode of using it, will destroy all ground of confidence in the 
result. So, in the science of mathematics, the meaning of the 
terms employed are accurately defined, while the language 
arising from the use of the symbols, in each branch, has a 
definite and precise signification. 



Whately's Logic, — of the operations of the mind and senses. 



INTKOD LOTION. II 

In the science of numbers, the ten characters, called figures, 
are the alphabet of the arithmetical language ; the combinations 
of these characters constitute the pure language of arithmetic; 
and the principles of numbers which are unfolded by means 
of this, ir, connection with our common language, constitute 
the science. 

If Geometry, the signs which are employed to indicate the 
boundaries and forms of portions of space, are simply the 
straight line and tlie curve; and these, in connection with our 
common language, make up the language of Geometry : a 
science which treats of space, by comparing portions of it 
with each other, for the purpose of pointing out their proper 
ties and mutual relations. 

Analysis is a general term embracing that entire portion of 
mathematical science in which the quantities considered are 
represented by letters of the alphabet, and the operations to 
be performed on them are indicated by signs. 

Algebra, which is a branch of Analysis, is also a species 
of universal arithmetic, in which letters and signs are employed 
to abridge and generalize all processes involving numbers. It 
13 divided into two parts, corresponding to the science and 
art of Arithmetic : 

1st. That which has for its object the investigation of the 
properties of numbers, embracing all the processes of reasoning, 
by which new properties are inferred from known ones ; and, 

2d. The solution of all problems or questions involving the 
determination of certain numbers which are unknown, from 
their connection with certain others which are known or given. 



12 ■ INTRODUCTION^. 

lu arithmetfc, all quantity is regarded as consisting of parts, 
which can be numbered exactly or approximatively, and la 
this respect, possesses all the properties of numbers. Proposi- 
tions, therefore, concerning numbers, have this remarkable pecu 
liarity, that they are propositions concerning all quantities 
whatever. Algebra extends the generalization still further. A 
number is a collection of things of the same kind, without refer- 
ence to the nature of the thing, and is generally expressed by 
figures. Algebraic symbols may stand for all numbers, or for all 
quantities which numbers represent, or even for quantities which 
cannot be exactly expressed numerically. 

In Geometry, each geometrical figure stands for a class; 
and when we have demonstrated a property of a figure, that 
property is considered proved for every figure of the class. In 
Algebra, all numbers, all lines, all surfaces, all solids, may be 
denoted by a single symbol, a or x. Hence, the conclusions 
deduced by means of those symbols are true of all things what- 
ever, and not like those of number and Geometry, true only 
for particular classes of things. The symbols of Algebra, there- 
fore, should not excise in our minds ideas of particular things. 
The wiitten characters, a, 6, c, J, x. y, 2, serve as the 
representatives of things in general, whether abstract or con- 
crete, whether known or unknown, whether finite or infinite. 

In the various uses which we make of these symbols, aid 
the processes of i-easoning carried on by means of them, tlie 
mind insensibly comes to regard them as things, and not as 
mere signs ; and we constantly predicate of them the properties 
of things in general, without pausing to inquire what kind of 



INTRODUCTION. IS 

thing is implied. All this we are at liberty to do, since the 
symbols being the representatives of quantity in general, there 
is no necessity- of keeping the idea of quantity continually alive 
in the mind; and the processes of thought may, without dan- 
ger, be allowed to rest on the symbols themselves, and there- 
fore, become to that extent, merely mechanical. But when we 
look back and see on what the reasoning is based, and how 
the processes have been conducted, we shall find that every 
step was taken on the supposition that we were actually 
dealing with things^ and not with symbols; and that without 
this understanding of the language, the whole system is without 
signification, and fails.* 

The quantities which are the subjects of the algebraic analysis 
may be divided into two classes: those which are known or 
given, and those which are unknown or sought. The known 
are uniformly represented by the first letters of the alphabet, 
«, i, c, c?, &c. ; and the unknown by the final letters, x^ y, 
2, V, &;c. 

Five operations, only, can be performed upon a quantity 
that will give results differing from the quantity itself: viz. 
1st. To add a quantity to it; 
2d. To subtract a quantity from it; 
3d. To multiply it by a quantity; 
4th. To divide it ; 
5th. To extract a root of it. 

Five signs only, are employed to denote these operations. 
They are too well known to be repeated here. These, with 

* Davies' Logic and Utility of Mathematics. § 278. 



14 INTKODUCTION. 

the signs of equality and inequality, together with the letters of 
fhe alphabet, are the elements of the algebraic language. 

The interpretation of the language of Algebra is the first 

thing to which the attention of a pupil should be directed; 

and he should be drilled in the meaning and import of the 

■•-mbols, until their significations and uses are as familiar as 

the sounds of the letters of the alphabet. 

All the apprehensions, or elementary ideas, are conveyed to 
the mind by means of definitions and arbitrary signs ; and 
every judgment is the result of a comparison of such impressions. 
Hence, the connection between the symbols and the ideas which 
-hey stand for, should be so close and intimate, that the one 
«hall always suggest the other; and thus, the processes of 
Algebra become chains of thought, in which each link tulfils the 
double ofiice of a distinct and connecting propos tion. 




LEMENTS OE ALGEBRA. 



CHAPTER I. 

DEFINITIONS AND PRELIMINARY REMARKS. 

1. Quantity is anything which can be increased or dimirz- 
Ished, and measured. 

2« Mathematics is the science which treats of the measurement 
and relations of quantities. 

3. Algebra is a branch of mathematics, in which the quantities 
considered are represented by letters, and the operations to be 
performed upon them are indicated by signs. The letters a?id 
signs are called symbols. 

4. In algebra two kinds of quantities are considered : 

Isi. Known quantities^ or those whose values are known or 
given. These are represented by the leading letters of the alpha- 
bet, as, «, 6, c, &c. 

2d. Unknown quantities^ or those whose values are not given. 
They are denoted by the final letters of the alphabet, as, 
X, y, 0, &c. 

r^etters employed to represent quantities are sometimes written 
with one or more dashes, as, a\ b", c"\ x\ ?/", &c., and are 
read, a prime^ b second^ c third, x prime, y second^ &c. 

5. The sign 4-, is called plus, and when placed between two 
quantities, indicates that the one on the right is to be added to 
the one on the left. Thus, a -\- b is read a plus b, and indicates 



16 ELEMENTS OF ALGEBRA. [CHAP. L 

that the quantity represented by h is to be added to the quan- 
tity represented by a. 

6. The sign — ^, is called minus, and when placed between two 
quantities, indicates that the one on the right is to be subtracted 
from the one on the left. Thus, c — c? is read c mingHB^^fd 
hidicates that the quantity represented by d is to be subtoacted 
from the quantity represented by c. 

Tlie sign -j-, is sometimes called the positive sign, and the 
quantity before which it is placed is said to be positive. 

The sign — , is called the negative sign, and quantities affected 
by it are said to be negative, 

7. The sign x , is called the sign of multiplication, and when 
placed between two quantities, indicates that the one on the left 
is to be multiplied by the one on the right. Thus, a X 6, iiidi 
cates that a is to be multiplied by b. The multiplication of 
quantities may also be indicated by placing a simple point 
between them, as a.h. which is read a multiplied by 6. 

Tlie multiplication of quantities, which are represented by 
letters, is generally indicated by simply writing the letters one 
after another, without interposing any sign. Thus, 
ab is the same as a x b, or a.b; 
and abc, the same as a X b X c or a.b.e. 

It is plain that the notation last explained cannot be employed 
when the quantities are represented by figures. For, if it were 
required to indicate that 5 was to be multiplied by 6, we 
c-uuld not write 5 6. without confounding the product with the 
number 56. 

The result of a multiplication is called the product, and each 
of the quantities employed, is called a factor. In the product 
of several letters, each single letter is called a literal factor. 
Thus, in the product ab there are two literal factors a and b ; in 
the product bed there are three, 6, c and d. 

8. The sign -i-, is called the sign of division, and when placed 
between two quantities, indicates that the one on the left is to be 
divided by the one on the right. Tlius, a — b indicates that o is to 



CHAP. I.] DEFINITIONS AND EEMARKS. 17 

be divided by h. Tlie same operation may be indicated by writing 
b under a, and drawing a line between them, as — ; or by writing 
b on the right of a, and drawing a line between them, as a\b. 

9. The sign =, is called the sign of equality^ and indicates that 
tno two quantities between which it is placed are equal to each 
other. Thus, a — b =. c -\- d^ indicates that a diminished by b is 
equal to c increased by d. 

10. The sign >, is called the sign of inequality^ and is used to 
indicate that one quantity is greater or less than another. 

Thus, a > 6 is read, a greater than b ; and a < ^ is read, a lest 
than b ; that is, the opening of the sign is turned toward the greater 

•quantity. 

11. The sign -^ is sometimes employed to indicate the difference 
)f two quantities when it is not known which is the greater. 

Thus, a r^ b, indicates the difference between a and b, without 
ohowing which is to be subtracted from the other. 

12. The sign ex, is used to indicate that one quantity varies as 

to another. Thus a oc -— , indicates that a varies as — -. 
6 6 

13. The signs : and : :, are called the signs of proportion; the 
first is read, is to, and the second is read, as. Thus, 

a : b : : c : d, 
is read, a is to b, as c is to d. 

The sign .♦., is read hence, or consequently. 

14. If a quantity is taken several times, as 

a-\-a-{-a-\-a-{-a^ 
it is generally written but once, and a number is then placed 
before it, to show how many times it is taken. Thus, 

a-\-a-{-a-\-a-\-a may be written 5a. 
The number -5 is called the co-efficient of a, and denotes that a is 
taken 5 times. 

Hence, a co-efficient is a number prefixed to a quantity, denoting 
the number of times which the quantity is taken. 

2 



18 ELEMENTS OF AL'IEBRA. [CHAP. L 

When no co-efficient is widtten, the co -efficient 1 is always under- 
stood; thus, a is the same as la. 

15. If a quantity is taken several times as a factor, the product 
may be expressed by writing the quantity once, and placing a 
number to the right and above it, to show how many times it is 
taken as a factor. 

Thus, aXaX€LXaxa may be written a*. 

Tlie number 5 is called an exponent, and indicates that a is 
taken 5 times as a factor. 

Hence, an exponent is a number written to the right and above 
a quantity, to show how many times it is taken as a factor. If 
no exponent is written, the exponent 1 is understood. Thus, a is 
the same as a^. 

16. If a quantity be taken any number of times as a factor, the 
resulting product is called a power of that quantity : the exponent 
d^iotes the degree of the power. For example, 

a^ = a is the fost power of a, 
©2 — a X a is the second power, or square of a, 
a? :=za X a X a is the third power, or cube of a, 
a^z=.aXaXoiXa is the fourth power of a, 
a^zraXaXaXaXais the fifth power of a, 
m which the exponents of the powers are, 1, 2, 3, 4 and 5 ; and 
the powers themselves, are the results of the multiplications. It 
should be observed that \}[i^ exponent of a power Ss, always greater 
by one than the number of multiplications. The exponent of a 
power of a quantity is sometimes, for the sake of brevity, called 
the exponent of the quantity. 

17. As an example of the use of the exponent in algebra, let 
it be required to express that a number a is to be multiplied 
tliree times by itself; that this product is then to be multiplied 
three times by 5, and this new product twice by c ; we should 
write 

axaxaxaxhxbxhxcxc = a'^¥c^. 
If it were ftirther required to take this result a cei-taiu number 
-of times, say seven, we should simply write la^b'^e' 



CHAP. I.] DEFINITIONS AND REMAEKS. 19 

18. A root of a quantity, is a quantity which being taken a 
certain number of times, as a ff^ctor, will produce the given 
quantity. 

The sign -/Tis called the radical sign, and when placed over 
a quantity, indicates that its" root is to be extracted. Thus, 
2/a or simply V^ denotes the square loot of a. 
l/a denotes the cube root of a. 
*/a denotes the fourth root of a. 
The number placed over the radical sign is called the index 
of the root. Thus, 2 is the index of the square root, 3 of tlia 
cube root, 4 of the fourth root, &c. 

19« Tlie reciprocal of a quantity, is 1 divided by that quantity. 
Thus, 

— is the reciprocal of a; 

and J- is the reciprocal of a-j- h. 

a-\-b 

20. Every quantity written in algebraic language, that is, by 
the aid of letters and signs, is called an algebraic quantity, or tlie 
algebraic expression of a quantity. Thus, 

(is the algebraic expression of three times the 

( quantity denoted by a ; 

( is the algebraic expression of five times the 

( square of a ; 

j is the algebraic expression of seven times the 

t product of the cube of a and the square of b ; 

_ _,, ( is the algebraic expression of the difference 
6a — oo\ ° . 

( between three times a and five times o; 

is the algebraic expression of tmce the square 

of a, diminished by three times the product 

of a and 5, augmented by four times the 

square of b. 

21. A single algebraic expression, not connected with any other 
by the sign of addition or subtraction, is called a monomial^ or 
simply, a term. 



2a2 - 3a^ -f 462 



20 ELEMENTS OF ALGEBRA. [CHAP, t 

Thus, 3a, 5a2, la^h^, are monomials, or single terms. 

An -algebraic expression composed of two or more terms cod. 
Elected by the sign + or — , is called a polynomial. 

For example, 3a — 55 and 2o? — Scb + 4:P, are polynomials. 

A polynomial of two terms, is called a binomial; and one of 
tlu'ee terms, a trinomial. 

22* The numerical value of an algebraic expression, is the num 
ber obtained by giving a particular value to each letter which 
enters it, and performing the operations indicated. This numer- 
ical value will depend on the particular values attributed to the 
letters, and will generally vary with them. 

For example, the numerical value of ^a^^ will be 54 if we make 
a = 3; for, 3^ = 3 X 3 X 3 =: 27, and 2 X 27 == 54. 

The numerical value of the same expression is 250 when wfi 
make a = 5; for, 53 = 5 X 5 X 5 = 125, and 2 x 125 = 250. 

We say that the numerical value of an algebraic expression 
generally varies with the values of the letters which enter it; it 
does not, however, always do so. Thus, in the expression a - b, 
so long as a and b are increased or diminished by the same 
number, the value of the expression will not be changed. 

For example, make a = 7 and 6 = 4: there results a — 5 = 3. 

Now, make a r= 7 + 5 = 12, and 6 = 4 + 5 = 9, and there 
results, as before, a — 6 = 12 — 9 = 3. 

23 • Of the different terms which compose a polynomial, some 
are preceded by the sign +, and others by the sign — . The 
former are called additive terms, the latter, subtractive terms. 

When the first term of a polynomial is plus, the sign is gene- 
rally omitted ; and when no sign is written before a term, it is 
always understood to have the sign -f-. 

24. The numerical value of a polynomial is not affected by 
changing the order of its terms, pro\ided the signs of all the 
lerms remain unchanged. For example, the polynomial 

4a-^ — 2a^ + 5ac2 = 5ac2 — SaH -\- 4a^ =: - Sa^ + 5ac2 + 4a^. 

25. Each literal factor which enters a term, is called a dimen- 
sion of the term ; and the degree of a term is indica.ted by the 
namber of these factors or dimensions. Thus, 



CHAP. I.] DEFINITION'S AND REMARKS. 2l 

3a is a term of one dimension, or of the first degree. 

6ab is a term of two dimensions, or of the second degree. 

la^c^ = laaabcc is of six dimensions, or of the sixth degree. 

In general, the degree of a term is determined by taking the sum 
of the exponents of the letters which enter it. For example^ the 
term Sa^bcd^ is of the seventh degree, since the sum of the expo- 
nents, 

2 + 1 + 1+3, is equal to 7. 

26* A polynomial is said to be homogeneous, when all of ita 
terms are of the same degree. The polynomial 

Sa — 26 + c is homogeneous and of the first degree. 

— 4ao + b^ is homogeneous and of the second degree. 

5a^c — 4:0^ + 2c'^d is homogeneous and of the third degree. 

Sa^ — 4ab +-c is not homogeneous. 

27» A vinculum , parenthesis ( ), brackets [], { }, or 

bar I, may be used to indicate that all the quantities which they 
connect are to be considered together. Thus, 
a-{- b -{- c X X, [a -\- b -{- c) X X, [a -\- b -\- c] X x, or {a -\~ b -\- c] x, 
indicate that the trinomial a + & + c is to be multiplied by x. 

When the parenthesis or brackets are used, the sign of mul- 
tiplication may be omitted : as, (a + 6 + c) a:. The bar is used 
in some cases, and differs from the vinculum in being placefl 
vertically, as + a x. 

+ 6 
+ c 

28. Terms which contain the same letters affected with equal 
exponents are said to be similar. Thus, in the polynomial, 

lab + Zab — 4a%^ + 5a%\ 
the terms 7ab and Sab, are similar, and so also are the terms 
— 4aW and 5aW, the letters .in each being the same, and thft 
same letters being affected with equal exponents. But in tho 

binomial 

Sa^ + 7a62, 

the terms are not similar; for, although they contain the same 
letters, yet the same letters are not affected with equal exp<i- 
nents. 



22 ELEMENTS OF ALGEBRA. LCHAP. I. 

29. When a polynomial contains similar jerms, it may be 
reduced to a simpler form by forming a single term from each 
set of similar terms. It is said to be in its simplest forrn^ when 
it contains the fewest terms to which it can be reduced. 

if we take the polynomial 

2a%c^ — 4a36c2 + Qa%c^ — Sa^c"^ + lla^c^, 
we know, from the definition of a co-efficient, that the literal 
part a'^bc'^ is to be taken additively, 2+6 + 11, or 19 tim(».s; 
and subtr actively, 4+8, or 12 times. 

Hence, the given polynomial reduces to 

It may happen that the co-efficient of the subtractive term, ob- 
tained. . as above, will exceed that of the additive term. In that 
case, subtract the positive co-efficient from the negative^ prefix the 
minus sign to the remainder, and then annex the literal part. 
Li the polynomial 

Za% + 2a26 - 5a26 - Sa^S, 
we have, + Sa^S — ^a% 

+ 2a26 — Za^b 



+ ba% —Sa^ 

But, — Sa'^b = — 5a'^b — Za^b : hence 

5a25 — Sa25 = ^a% — ba% — Sa^ = — Sa^. 

In like manner we may reduce the similar terms of any poly- 
Bomial. Hence, for the reduction of a polynomial containing 
sets of similar terms, to its simplest form, we have the following 

EULE. 

I. Add together the co-efficients of all the additive terms of each set^ 
and annex to their sum the literal part : form a single subtractive 
term in the saine manner. 

II. Then^ subtract the less co-efficient from the greater^ and to the 
remainder prefix the sign of the greater co-efficient^ and annex th$ 
literal part. 



CHAP. I.] EEDUCTION OF POLYNOMIALS. 23 

EXAMPLES. 

1. Reduce the polynomial 4a26 — Qa^b — 9a^b + iia^b lo its 
simplest form. Ans. — 2a'^b. 

2. Reduce the polynomial labc^ — abc"^ — 7abc'^ — Sabc'^ + Qabc^ 
to its simplest form. Ans. — oabc^. 

3. Reduce the polynomial 9cb^ — Sac^ + \bc¥ + Sea -f 9ac^ 

— 24c63 iQ i^g simplest form. • Ans. ac^ + 8ca. 

4. Reduce the polynomial Qac^ — 5a6^ + lac^ — 3a6^ — 13ac* 
•I- IStzi^ to its simplest form. ' Ans. l^ab^. 

5. Reduce, the polynomial abc^ — ubc + Sac^ — ^abc^ + ^^^ 

— 8ac2 to its simplest form. Ans. — Sabc"^ + 5a6c — Bac^. 

6. Reduce the polynomial 3^262 _ ^a^ + bab - 9a%^ + Oa^Z^ 
+ Sab to its simplest form. Ans. — Qa^^b"^ -f- 2aH + 8^^* 

7. Reduce the polynomial Sacb^ — lahW — 6a*6^ — Sacb^ 
-\ 6ahW — 6acb'^ + 4a'*^^ -f 2a^b^ to its simplest form. 

Ans. — a^c^^^ — Qacb'^. 

8. Reduce the polynomial — la?b'^c'^ 4- ^a^bc"^ + Ga^S^c^ -j- a^^^c^ 

— 5a^bc^ — b^c^ to its simplest form. Ans. 4a^bc^ — b^c^. 
.9. Reduce the polynomial — lOa^^ -{- Qa^P -\- 7a?b — 5a^^ 

— 6a^ + Sa^b"^ to its simplest form. Ans. — Sa^ + 4^262. 

Remark. — It should be observed that the reduction affects only 
the co-efficients, and not the exponents. 

30. A THEOREM is a general truth, which is made evident by a 
course of reasoning called a demonstration. 

A PROBLEM is a question proposed which requires a solution. 

31. We shall now illustrate the utility and brevity of algebraic 
language by solving the following 

PROBLEM. 

The sum of hoo numbers is 67, and their difference is 19 ; what 
are the numbers ? 

Let us first indicate, by the aid of algebraic symbols, the 
relation which exists betweer: the given and unknown numbers 
of the problem. 



24 ELEMENTS OF ALGEBRA. [CHAP, i 

If the less of the two numbers were known ^ the greater could 

he found by adding to it the difference 19 ; or in other words, 

the less number, plus 19, is equal to the greater. 

If, then, we denote the less number by x^ 

re + 19 will denote the greater, 

and 2a; + 19 will denote the sum. 

But from the enunciation, this sum is to be equal to 67. Thero 

iJ>re. 

2;r + 19 = 67. 

Now, if '\lx augmented by 19, is equal to 67, 2a: alone is equal 

fc() 67 minus 19, or 

2a: = 67 - 19, 

or performing the subtraction, 

2a; = 48. 

Hence, x is equal to half of 48, that is 

48 ^, 
. = - = 24. 

The less number being 24, the greater is 
a; + 19 = 24 + 19 = 43. 
And, indeed, we have 

43 + 24 = 67, and 43 - 24 = 19. 

GENERAL SOLUTION. 

The sum of two numbers is a, and their difference is h Wlial 
are the two numbers ? 

Let X denote the less number ; 

Then will x -\-h denote the greater number. 

Now, from the conditions of the problem, 
a; + a; -f 6, or 2x -\- b 
'A ill be equal to the sum of the two numbers : hence, 

2x^-b = a. 
Now, if 2a; + 6 is equal to a, 2a; alone must be equal U 
a — b and 

__ a — i _ a b 
"^ ~ ~2~ ~ 2" "" "2 • 



CHAP. I.J SOLUTION OF PROBLEMS. 25 

If the value of x be increased by &, we shall have the 
greater number : that is, 

a b ah 

^ 2 2 ^^ 2^2' 

he»ce, X -\- h ^= --■ -{- -— =z the greater number, and 

X =z- — = the less number. 

That is, the greater of two numbers is equal to half their sum 
increased by half their difference ; and the less is equal to half 
their sum diminished by half their difference. 

As the form of these results is independent of any particular 
values attributed to the letters a and ^, the expressions are called 
formulas^ and may be regarded as comprehending the solution 
of all problems of the same kind, differing only in the numerical 
values of the given quantities. Hence, 

A formula is the algebraic expression of a general rule, or 
principle. 

To apply these formulas to the case in which the sum is 237 
and difference 99, we have 

336 

the greater number = ^^^ -f- ^ = ^' ^ — = — — = 168 ; 



237 99 


237 + 99 


'"^ - 2 + 2 - 
237 99 


2 
237 - 99 


^ 2 2 " 


2 



1 OQ 

and the less ^ == iri. _ ^ =, ^^;^__^ = -^ "= ^^'^ 

and these are the true numbers ; for, 

168 + 69 =; 237 which is the given sum, 
and 168 — 69 = 99 which is the given differenoa. 



CHAPTER n. 

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. 



ADDITION. 

31 i Addition, in algebra, is the operation of finding the sim- 
plest equivalent expression for the aggregate of two or more alge- 
braic quantities. Such equivalent expression is called their sum. 

32i If the quantities to be added are dissimilar, no reductions 
can be made among the terms. We then write them one 
after the other, each with its proper sign, and the resulting 
polynomial will be the simplest expression for the sum. 

For example, let it be required to add together the mono- 
mials 

3a, 55 and 2c ; 
we connect them by the sign of addition, 

* 3a + 56 + 2c, 

a result which camiot be reduced to a simpler form. 

33. If some of the quantities to be added have similar terms, 
we coimect the quantities by the sign of addition as before, 
and then reduce the resulting polynomial to its simplest form, 
by the rule already given. This reduction will, in general, be 
more readily accomplished if we write down the quantities to 
be added, so that similar terms shall fall in the same column. 
Thus; 

X . -, . -, /. -. 1 /. r 3a2 — Aab 

Let it be required to find the sum of \ 

. . ^ -^ 2a2 — Sa5 + Z»2 

the quantities, i 

^«^ - 55^ 
Their sum, after reducing (Art. 29), is - ba^ — bab ~ 46^ 



GHAP II.] 



ADDITION. 27 



34. As operations similar to the above apply to all algebraic 
expressions, we deduce, for the addition of algebraic quantities, 
the following general 

EULE. 

I. Write down the quantities to be added, with their respective 
signs, so that the similar terms shall fall in the same column. 

II. Reduce the similar terms, and annex to the results those ternih 
which cannot be reduced, giving to each term its respective sign. 



3^2 _ 4j^5 _ 21^2 
5«2 + 2ub - &2 
+ S^b - 2&2 
8a2 -{. ab - 562 _ 3^2 



EXAMPLES, 

1. Add together the polynomials, 

3a2_262_4a6, 5a2 - ^2 + 2a6 and 3a6 - 3c2 - 262. 

The term 3a2 being similar to 5a2 
we write Sa^ for the result of the re- 
duction of these two terms, at the same ■{ 
time slightly crossing them as in the 
terms of the example. 

Passing then to the term — 4a6, which is similar to the two 
terms + 2a6 and + 3a6, the three reduce to + a6, which is 
placed after 8a2, and the terms crossed like the first term. 
Passing then to the terms involving 62, we find their sum to be 
— 562, after which we write — 3c2. 

The marks are drawn across the terms, that none of them 
may be overlooked and omitted. 



Sum 



Sum 



(2). 

7rr4-3a6+ 2c 
^Sx-Sab- 5c 

5x — 9a6— 9c 
. 9.r-9a6-12c 




(3). 

16a262+ bc-2abc 

- 4a262 - 96c + Qabc 

-9a262+ 6c + abc 
3a2j2 _ 75c _^ 5^7,^ 


(4). 
a-\- ab — cd-{- f 
3a + 5a6-6c(^- / 

— 5a - 6a6 + Qcd — 7/ 

- a+ a6+ cd + 4:f 


(5). 

6a6 + cd^ d 

Sab + 5cc? — y 

-4a6+ Qcd + x 

— hah — I2cd + y 


— 2a 4- a6 + - 


-3/ 


-{-x-hd 



28 ELEMENTS OF ALGEBRA. [CHAP. IL 

6. Add together 3a + b, 3a + 36, — 9a — 7b, Qa + 96 and 
8a + 36 -I- 8c. Ans. 11a + 96 +8c. 

7. Add together Sax 4 3ac +;^ — 9ax -f 7a + c?, + 6az f 3a£ 
-{- 3/, Sax + 13ac + 9/ aiid — 14/+ 3aa;. 

^ns. llaa; + 19ac — /+ 7a 4- c?. 

8. Add together the polynomials, Sa'^c + 5a6, 7a2c — 3a6 + Sae 
Sa^c — 6a6 |- 9ac and — Sa^c + a6 — 12ac. Ans. laP-c — 3a6. 

9. Add the polynomials, \9a?x'^b — 12a3c6, ha?x'^b + Ihahb 

— ]Oaa;, — ^a?x^b — \Zahb and — 18a2a:36 — 12a3c6 + 9aa;. 

Ans. 4:0? x^b — 22a^cb — ax. 

10. Add together 3a + 6 + c, 5a + 26 + 3ac, a -{- c -{- ac and 

— 3a — 9ac — 86. Ans. 6a — 56 -f 2c — 5ac. 

11. Add together 5a26 + 6ca; + 96c2, 7c:c — 8a26 and — 15c« 

— 96c2 + 2a26. ^^5. — a26 — 2ca;. 

12. Add together 8aa; + 5a6 + 3a26V, - 18aa; + 6a2 + lOaS 
and lOaa; — 15a6 — 6a262c2. Ans. —3aWc^-\-6a'^. 

13. What is the sum of 4AaWc — 21abc —Ma'^y and 10a36'-^r 
+ 9a6c'? Ans. blaWc — 18a6c -14aY 

14. What is the sum of 18a6c — 9a6 + Gc^ — 3c + 9ax and 
9a6c + 3c — 9aa; ? Ans. 21fabc — 9ab 4- Gc"^. 

15. What is the sum of 8a6c 4- b^a — 2cx — 6xt/ and 7ca 

— xy — 1363a % Ans. Sabc — 12b^a 4- 5ca: — 7xy. 

16. ^Yhat is the sum of 9a2c — 14a6y 4- 15a262 and — a^c 
— 8a262? Ans. Sa\ ~l4.aby ^la^"^. 

17. What is the sum of 17a*62 + 9a36 — 3a2, — 14a562 4- la^ 

— 9a3, — 15a36 4- 7a562 — a^ and 14a36 — 19a36? 

Ans. , 

18. What is the sum of Say* — 9ax^ — 17 axy, 4- 9a^2 .j_ jg^^j 
4- Maxy and 70^6 4- 3a.'c3 — 7arc2 4- 46ca; ? Ans. . 

19. Add together 3a2 4- 5a262c2 ^ 9a3^, 7a2 — 8a262c2 — lOa-^a? 
and 10a6 + 16a262c2 4- 19a^x. Ans. lOa^ 4- 13a262c2 4- 10a6. 

20. Add together 7a26 — 3a6c — 862c — 9c3 4- cd% Scbc — 5a^ 
+ 3c3 — 462c 4- cd^, and 4a26 — 8c3 4- 962c — ScP. 

Ans. 6a^ + 5abc - 362c -14c3 4- 2c(P - Scl^ 



CHAP. II.J SUBTRACTION". 29 

21; Add together — ISa^ + 2a¥ -f- Qa^^, — Sab^ + 7^36 - 6a^^ 
and — 6aH -f 6a6* ^p lla^'^. Ans. — 16a?6 + 12a^P. 

22. What is the sum of Sii^^Zc _ lea^^; _ 9axM, + Ga^^^Zc 
- Qax^d + .7a% and + IQax^d — a*^ — 8a362c 1 

23. What is the sum of the following terms : viz., 8a^ — 10a*6 
~ 16c%^ .+ 4a^3 — 12a*6 + 15a^^ + 24a2Z,3 _ ga^* — IGa^i^ 
4 20^253 _^32a6*- 865? 

Ans. 8a5 - 22^46 - lla^b^ + 48a263 + 26a64 — 86*. 



SUBTRACTION 

35i Subtraction, in algebra, is the operation for finding the 
simplest expression for the difference between two algebraic 
vjuantities. This difference is called the remainder. 

36. Let it be required to subtract 46 from 5a. Here, as 
ifie quantities are not similar, their difference can only be indi- 
cated, and we write 

5a - 46. 

Again, let it be required to subtract 4a36 from 7a^. These 
fcerms being similar, one of them may be taken from the other 
and their true difference is expressed by 
^a% - 4a36 =:= 3a36. 

37« Generally, if from one polynomial we wish to subtract 
another, the operation may be indicated by enclosing the second 
in a parenthesis, prefixing the minus sign, and then writing it 
after the first. To deduce a rule for performing the operation 
thus indicated, let us represent the sum of all the terms in the 
first polynomial by a. Let c represent the sum of all the ad- 
ditive terms in the other polynomial, and — d the sum of 
the subtractive terms ; then this polynomial will be represented 
by c — d The operation may then be indicated thus, 

a — [c — d) ; 
where it is required to subtract from a the difference between 
( and d. 



30 ELEMENTS OF ALGEBRA. ^CHAP. II. 

If, now, we diminish the quantity a by the quantity c, the 

result a — c will be too smal by the quantity d, since c should 

have been diminished by d before taking it from a. Hence, 

to obtain the true remainder, we must increase the first result 

by d, which gives the expression 

a — c-\- d, 
and this is the true remainder. 

By comparing this remainder with the given polynomials, we 
see that we have changed the signs of all the terms of the quantity 
to be subtracted, and added the result to the other quantity. To 
facilitate the operation, similar quantities are written in the same 
column. 

Hence, for the subtraction of algebraic quantities, we have the 
following 

RULE. 

I. Write the quantity to be subtracted under that from which it 
is to be taken, placing the similar terms, if there are any, in the 
same column. 

U. Change the signs of all the terms of the quantity to be sub- 
tracted, or conceive them to be changed, and then add the result to 
the other quantity. 

EXAMPLES. 

(1). ■ fit (1). 

From - Qac — 5a6 + c^ | ° J Qac — oab -f- c^ 

Take - Sac -\- Sab — 7c ".|| — Sac — Sab + 7c 

Remainder Sac — Sab 4-^2 + 7c. II | ' Sac - Sab -\- c^ + 7c, 

5 o — ' 

(3). " (3). 

From - 16a2 — bbc -\- 7ac I9abc — Wax — 5axif 

Take • Ua^ + 56c + Sac 17abc-\- 7ax — 16axy 

Remamder 2a'^ — 106c — ac 2abc — 2Sax + lOaxy 

(4). (5). 

From - 5a3_4a26 4- Sb^c Aab — cd\-Sa'^ 

Take - - 2a3 + 3a26- 86^0 ^ab - 4cd + Sa^ -}- 55^ 

Remamder 7a^ ~7a'^b + Ub^c -^ ab + Scd -{- — 5b\ 



CHAP. II.] SUBTEACTION. 31 

6. From Sa^x — ISabc + 7a2, take 9a^x — ISabc. 

Ans. — 6a'^x + 7a^. 

7. From 5U-b^c — 18a6c — Ua^i/, take 41aWc — 27abc 

- 14a2y.' ylns. lOa^^^g + 9abc. 

8. FBom 27a6c — 9ab + 6c2, take 9abc + 3c — 9aa;. 

^ws. 18a6c — 9ab + 6c2 — 3c + ^x. . 

9. From 8a6c — 1263a + Sea; — 7xy^ take 7ca; — xy — 136%, 

^;is. 8a6c + 6% — 2cx — 6a*y. 

10. From Sa^c — 14a6y + 7a262, take 9a2c - I4.aby + \^a?b\ 

Ans. — a^c — 'Qa%'^. 

11. From 9a6a;2 — 13 + '^^ab'^x — 4:b^cx^, take 366ca;2 + 9a^x^ 

- 6 + 3a63a;. ^n5. llab^x — 7bhx^ - 7. 

12. From 5a* - 7a362 - 3c5c^2 + 7^^ take 3a* - 3a2 - 7c^d^ 

- 15a362. Ans. 2a* + 8a362 + 4c^d^ -}- 7d + Sa^ 

13. From 51a262 _ ^Sa^ + 10a*, tal^e 10a* - Sa% — 6aW. 

Ans. h7aW — 40a%. 

14. From 21x^y^ + 25x^y^ + 6Sxy^ — 40y^, take 64a;2yS 
4- 48a;?/* — 402/5. ^^5, 20:cy* — S9x^y^ + 21a;V- 

15. From 5Sx^y^ — 16x'^y^ — ISx^y — 5Qx^, take — 15a;2y3 
+ 18a;3y2 + 24a;*?/. Ans. S5x^y^ — 42a;*?/ — 56a;5. 

38. From what has preceded, we see that polynomials may be 
subjected to certain transformations. 

For example - - - - 6a2 — 3a6 + 262 _ 26c, 
may be written - . - . 6a2 — (3a6 — 262 _|_ 26c). 
In like manner ... - 7a^ — 8a26 — 462c + 66^, 
may be written . . . . 7a3 — (8a26 4- 462c — 663) . 

or, again, 7a3 — 8a26 — (462c — 663). 

Also, 8a2 — 6a262 + 5a263, 

■becomes 8a2 — (6a262 — 5a263). 

Also, - 9a2c3 — 8a* + 62 — c, 

may be written . - - . 9a2c3 — (8a* — 62 + c) ; 
or, it may be written - - 9a2c3 + 62 — (8a* + c). 

These transformations consist in separating a polynomial into 
two parts, and then connecting the parts by the minus sign. 



32 ELEMENTS OF ALGEBRA. [CHAP. IL 

It will be observed that the sign of each term is changed when 
the term is placed within the parenthesis. Hence, if we have 
one or more terms included within a parenthesis having the 
minus sign before it, the signs of all the terms must be changed 
when the parenthesis is omitted. 

Tlius. 4a — (Qab - 3c — 2b), 

is equal to 4a — 6ab + 3c + 26. 

Also, Qab — (— 4:ac -{- Sd — 4a6), 

is equal to 6ab + 4ac — Sd -{- 4ab. 

39. Remark. — From what has been shown in addition and 
subtraction, we deduce the following principles. 

1st. In Algebra, the words add and sum do not always, as in 
arithmetic, convey the idea of augmentation. For, if to a we add 
— 5, tlie sum is expressed by a — b, and this is, properly speaking, 
the arithmetical difference between the number of units expressed 
by a, and the number of units expressed by b. Consequently, 
this result is actually less than a. 

To distinguish this sum from an arithmetical sum, it is called 
the algebraic sum. 

Thus, the polynomial, 2a^ — SaM + 362c, 
is an algebraic sum, so long as it is considered as the result of 
the union of the monomials 

2a3, — 3a26, + 362c, 
with their respective signs; but, in its proper acceptation, it is 
the arithmetical difference between the sum of the units con- 
tained in the additive terms, and the units- contained in the 
subtractive term. 

It follows from this, that an algebraic sum may, in the numer 
leal applications, be reduced to a negative expression. 

2d. The words subtraction and difference, do not always convey 

the idea of diminution. For, the difference between -fa and 

— 6 being 

a — ( — 6) = a-}-6, 

is numerically greater than a. This result is an algebraic differ 
ence. 



CHAP. II.] MULTIPLICATION 33 

40i It frequently occurs in Algebra, that the algebraic sign -f- 
or — , which is written, is not the true sign of the term before 
which it is placed. Thus, if it were required to subtract — h 
from a, we should write 

a — ( — h) ^^ a -{- b. 
Here the true sign of the second term of the binomial is plus, 
although its algebraic sign is — . This minus sign, operating 
upon the sign of 6, which is also negative, produces a plus sign 
for b in the result. The sign which results, after combining the 
algebraic sign with the sign of the quantity, is called the esset^ 
Hal sign of the term, and is often different fromi the algebraic 
sign. 



MULTIPLICATION-. 

41. Multiplication, in Algebra, is the operation of finding tlie 
product of two algebraic quantities. The quantity to be multi- 
plied is called the multiplicand ; the quantity by which it is 
multiplied is called the multiplHer ; and both are called factors. 

42« Let us first consider the case in which both factors are 
monomials. 

Let it be required to multiply Ta^i^ by 4a25 ; the operation 
may be indicated thus, 

IfaW X 4.a^, 
or by resolving both multiplicand and multiplier into their 
simple factors, 

laaabb X 4aa6. 
Now, it has been sho\\Ti in arithmetic, that the value of a 
product is not changed by changing the order of .ts factors ; 
hence, we may write the product as follows: 

7 X 4aaaaabbb, which is equivalent to 28a^b^. 
Comparing this result with the given factors, we see that the 
co-efficient in the product is equal to the product of the co-effV- 
cients of the multiplicand and multiplier ; and that the exponent 
of each letter is equal to the sum of the exponents of that letter 
in both multiplicand and multiplier. 

3 



84 



ELEMENTS OF ALGEBRA. 



[CHAP. 11. 



And since the same course of reasoning may be applied to 
any two monomials, we haye, for the multiplication of mono 
mials, the following 

RULE. 

J. Multiply the co-efficients together for a new co-efficient. 

II. Write after this co-efficient all the letters which enter into the 
multiplicand and multiplier^ giving to each an exponent equal to 
the sum of its exponents in both factors. 

EXAMPLES. 

(1) - - 8a25c2 X lahd"^ z=z ^aWc^d"^. 

(2) - - 2laWdc X 8a6c3 = 168a*&Vc?. 



Multiply- 



(3) 
Za% 

25a2 



(4) 
- I2a^x 



14Aa'^x^7/ 



(5) 
Qxgz 

ag^z 

6axg^z^ 



(6) 

a^xi 
2xy^ 



a^xg 



2a'^xY' 



Ans. b^a^^lP c'-d. 

Ans. QOabcd^. 

Ans. 7a^b^d^c*. 



7. Multiply Sa^h'^c by 7a%^cd. 

8. Multiply 5a&cZ3 by l2cdK 

9. Multiply 7a^hdh'^ by ahdc. 

43* We will now proceed to the multiplication of polynomials. 
in order to explain the most general case, we will suppose the 
multiplicand and multiplier each to contain additive and sub- 
tractive terms. 

Let a represent the sum of all the additive terms of the multi- 
plicand, and — h the sum of the subtractive terms ; c the sum 
•of the additive terms of the multiplier, and — d the sum of 
the subtractive terms. The multiplicand will then be represented 
by a — b and the multiplier, by c — d. 

We will now show how the multiplication expressed bv 
{a — b) X (c — cZ) can be effected. 

The required product is equal to a — 6 
taken as many times as there are units 
in c — d. Let us first multiply by c ; 
that is, take a — b as many times as 
there are units in c. We begin by writ- 
ing ac, which is too great by b taken 



a — b 
c — d 
ac — be 

— ad -{- bd 
ac — be — ad -{- hd. 



CHAP. II.] MULTIPLICATION. 35 

c times ; for it is only the difference between a and 5, that is 
first to be multiplied by c. Hence, ac — he is the product of 
a — h by c. 

But the true product is a — h taken c — d times : hence, the 
last product is too great by a — 6 taken d times ; that is, by 
ad — 6c?, which must, therefore, be subtracted. Suotracting this- 
from the first product (Art. 37), ws have 

{a — h) X {c — d) ^^ ac -- he — ad •\- hd '. 

If we suppose a and c each equal to 0, the product will re 
duce to i- hd. 

44. By considering the product of a — h by c — c?, we may 
deduce the following rule for signs, in multiplication. 

When two terms of the multiplicand and multiplier are affected 
with the same sign, their product will he affected with the sign -f-^ 
and when they are affected with contrary signs, their product wiU 
he effected with the sign — . 

We say, in algebraic language, that -f multiplied by + 
or — multiplied by — , gives + ; — multiplied by +, or + nau] 
tiplied by — , gives — . But since mere signs cannot be multi 
plied together, this last enunciation does not, in itself, express a 
distinct idea, and should only be considered as an abbreviation 
of the preceding. 

This is not the only case in which algebraists, for the sake of 
brevity, employ expressions in a technical sense in order to se- 
cure the advantage of fixing the rules in the memory. 

45. We have, then, for the multiplication of polynomials, the 
following 

RULE. 

MultiiAy all the terms of the multiplicand hy each term of the 
rmiltiplier in succession, affecting the product of any two terms with 
(he sign plus, when their signs are alike, and with the sign minus^ 
when their signs are unlike, TJien redvrie the polynomial result 
to its simplest form. 



ELEMENTS OF ALGEBRA. I CHAP. U, 



EXAMPLES. 



I. Multiply . - - - - - - - 3a2 + 4a6 -f- 63 

by - 2a + 5& 

6a^ 4- Sa^ + 2ab'^ 

+ 15a^ + 20ab^ + 5&» 
Product I- - - 6a3 -f 2^a^ + 22a62 + 56^ 

(2). (3). 

^2 _|_ y2 x^ -{- xy^ + 7aa; 

X — y ax -\- 5ax 

«3 -^ a;2/2 ax^ + aa;^^^ + Ta^a;^ 

— x^y — y^ 4- 5«^^ + hax^y^ 4- 35a^a;' 

a:^ -|- a;y2 — a-'^y — y^ Gax-^ -f- Qcix'^y^ -\- 42a'^^^. 

4. Multiply a;2 + 2ax -{•a?' by a; + a. 

Ans. x"^ + Saa:^ -f Sa^a; 4" «^. 

5. Multiply a;2 4- 2/2 by a; 4- 3^- 

^715. x"^ 4" ^2/^ 4" ^^y + y^. 

6. Multiply 3a62 4- QaH'2' by 3a&2 4- SaV. 

^/i5. 9a26* 4- 27aWc^ + 18a*c*. 

7. Multiply 4a;2 — 2y by' 2?/. ^?i5. S.'c^y — 4y\ 

8. Multiply 2a; 4- 4y by 2a; — 4y. Ans. 4x^ — 16y2. 

9. Multiply x^ 4- a;^?/ 4- ^y"^ + y^ by a: — y, Ans. . 

10. Multiply a;2 4- a;y 4- 3/^ by x"^ — a;y + 2/^. 

^ns. x^ 4- a;2y2 _|_ ^4^ 

In order to bring together the similar terms, in the product o 
tw'o polynomials, we arrange the terms of each polynomial Tv4tn 
reference to a particular letter ; that is, we arrange them so tha 
the exponents of that letter shall go on diminishing from left 
to right. 

11 Multiply 4a3 - 6a^ - Sab^ + 2b^ 

by 2a2- Sab - 4h^ 

Sa^ — lOa'^b — 26aW -\- 4aH^ 

— 12a*6 4- 15a-'^62 4- 24a^^ — 6ab* 

- Wa^^ 4- 20a2&3 _^ 32^54 __ q^ 

8a5 - 22a*6 - 17aW + 4Sa^^ 4" 2Qab^ — 86*. 



CHAP. II.J MULTIPLICATION. 37 

After having arranged the polynomials, with reference to the 
letter a, multiply each term of the first, by the term 2a^ of the 
second ; this gives the polynomial SaP — 10a*6 — l^aW -f- ^a^js, 
in which the signs of the terms are * the same as in the multi- 
plicand. Passing then to the term — Sab of the multiplier, mul- 
tiply each term of the multiplicand by it, and as it is affected 
with the sign — , affect each product with a sign contrary t<i 
that of the corresponding term in the multiplicand ; this gives 

— 12a*6 + l5aW + 24a2Z*3 _ Qab^^ 
Multiplying the multiplicand by — 4:b^, gives 

- IQaW + 20a^^ + S2ab^ — S¥. 
The product is then reduced, and we finally obtain, for the most 
simple expression of the product, 

8a5 — 22a*5 — 17a^^ + ASa^^ + 26a5* - Sb^ 

12. Multiply 2a2 — Sax + 4a;2 by 5a2 — 6ax — 2x^. 

Ans. 10a* — 27a^x + S4ca'^x'^ — ISax^ - 8xK 

13. Multiply Sx'^-2i/x + 6 by x^-\-2xy — S. 

Ans. Sx* + 4:X^y — 4:X^ — 4:X^y^ + IQxy — 15. 

i4. Multiply Sx^ + 2x'^y^ -f Sy^ by 2x^ - Sx^ -f 5^/3. 

15. Multiply Sax — 6ab — c by 2ax + a6 + e. 

Ans. 16a2a;2 — 4a^x — QaW + Qacx — labc — c^, 

16. Slultiply 3a2 - 6b^ + 3c2 by a^ - b\ 

Ans. 3a* - 5a262 + 3a2c2 - 3a263 ^ 555 _ 3j3c3. 

17. Multiply 3a2 - bbd + cf 
by _ 5^2 _^ 45^ _ 8c/: 

Product - 15ft* + S7a%d - 29a^/' - 2052a-2 + 44bcdf— StlA 

18 Multiply 4a>^62 — 5a262c + 8a26c2 - 3a2c3 — 7a6c3 
by 2a52 -4a6c — 26c2 + c^. 

8a*6* -- 10a36*c + 28a363c2 - 34a ^^2^3 
Product -^ — 4a263c3 — Ua^bh + 12a36c* + 7a262c4 , 
4- 14a26c5 + 14a62c5 — 3a2c6 - 7abc^ 



Qx^ — ^xY — 6a:V + 6a;33/2 + 15a;V' 



88 ELEMENTS OF ALGEBRA. [CHAP. IL 

46 • REMARKS ON THE MULTIPLICATION OF POLYNOMIALS. 

IsL If both multiplicand and multiplier are homogeneous, the 
product will he homogeneous, and the degree of any term of the 
product will be indicated by the sum of the numbers which indicate 
the degrees of its two factors. 

Thus, in example 18th, each term of the multiplicand is of 
the 5th degree, and each term of the multiplier of the 3d de- 
gree : hence, each term of the product is of the 8th degree. 
This remark serves to discover any errors in the addition of 
t^e exponents. 

2c?. If no two ter?7is of the product are similar, there will be no 
reduction a7nongst them ; and the number of terms in the product 
will then be equal to the number of terms in the multiplicand, multi 
plied by the number of terms in the multiplier. 

This is evident, since each term of the multiplier will produce 
9S many terms as there are terms in the multiplicand. Thus, in 
example 16th, there are three terms in the multiplicand and two 
m the multiplier : hence, the number of terms in the p-'o^luct is 
equal to 3x2 = 6. 

3rf. Among the terms of the product there are always two which 
cannot be reduced with any others. 

For, let us consider the product with reference to any letter 
common to the multiplicand and multiplier: Then the irreduci- 
ble terms are, 

1st. The term produced by the multiplication of the two terms 
ef the multiplicand and multiplier which contain the highest 
power of this letter ; and ^ 

2d. The term produced by the multiplication of the two terms 
which contain the lowest power of this letter. 

For, these two partial products will contain this letter, to a 
higher and to a lower power than either of the other partial pro 
ducts, and consequently, they cannot be similar to any of them. 
This remark, the truth of which is deduced from the law of 
rift exponents, will be very useful in division. 



..'ii.- 



CHAP. IIO 


MULTIPLICATION. 




EXAMPLE. 


Multiply - 


- 5a*62 + Sa^ — ab* — 2ab^ 


by . - 


a% — a62 


Product 





89 



If we examine the multiplicand and multiplier, with reference 
to a, we see that the product of 5a*62 by a^, must be irre- 
ducible ; also, the product of — 2ab^ by ab'^. If we consider 
the letter 6, we see that the product of — a6* by — ab"^, must 
be irreducible, also that of Sa% by a^b. 

47 1 The folloAving formulas depending upon the rule for mul- 
tiplication, will be found useful in the practical operations of 
algebra. 

Let a and b represent any two quantities ; then a -}- b will 
represent their sum, and a — b their difference. 

I. We have {a -j- by = [a + b) X {a -{- b), 
or performing the multiplication indicated, 

{a + by -a^ + 2ab + b^ ; that is, 
The square of the- sum of tivo quantities is equal to the square 
of the first, plus twice the product of the first by the second, plus 
the square of the seco7id. 

To apply this formula to finding the square of the binomial 
5a2 + 8a25, 
we have {ha^ -f ^a^by = 25a> + 80a*6 -{- 64a462. 
Also, {Qa^b + 9a63)2 ~ Z^a^"^ + lOSa^i* + ^laW, 

II. We have, {a — by = {a — b) x {a ^ 5), 
or performing the multiplication indicated, 

{a - by = a''- 2ab -f- b^ ; that is. 
The square of the difference between two quantities is equal to 
the square of the first, minus twice the product of the first by tin 
second, plus the square of the second. 

To apply this to an example, we have 

(7a2.52 _ [2ab^y = 4Qa^b^ - IGSa^Js + lAAa'^b^. 
Also, (4a363 7cV3)2 z= Ua%^ - b^a^^H^ + 49c*rf«. 



40 ELEMENTS OF ALGEBRA. LCHAP. IL 

III. We^have {a-\- b) X (a - b) = a^ - b\ 
by performing the multiplication ; that is, 

The sum of two quantities multiplied by their difference is equ-ul 
U) the difference of their squares. 

To apply tliis formula to an example, we have 

(8a3 + 7a62) x {Sa^ - lab^) = 64a6 _ 49^254. 

48. By considering the last three results, it is perceived 
tuat their composition, or the manner in which they are formed 
from the multiplicand and multiplier, is entirely mdependent of 
any particular values that may be attributed to the letters a and 
b, which enter the two factors. 

The manner in which an algebraic product is formed from its 
two factors, is called the law of the product ; and this law re- 
mains always the same, whatever values may be attributed to 
the letters wliich enter into the two factors. 



DIVISION. 

49. Divisiox, in algebra, is the operation for finding from two 
given quantities, a third quantity, which multiplied by the second 
shall produce the first. 

The first quantity is called the dividend, the second, the divisor^ 
and the third, or the quantity sought, the quotient. 

50. It was shown in multiplication that the product #f two 
terms having the same sign, must have the sign +, and that 
the product of two terms having unlike signs must have the 
sign — . Now, since the quotient must have such a sign that 
when multiplied by the divisor the product will have the sign of 
the dividend, we nave the following rule for signs in division. 

If the dividend is + and the divisor -p the quotient is -f- 
if the dividend is + and the divisor — the quotient is — 
if the dividend is — and the divisor + the quotient is — 
if the dividend is — and the divisor — the quotient is -f • 

That is : The quotient of terms having like signs is plus, ana 
iJie quotient of terms having unlike signs is mini-.s. 



CHAP. II.J DIYISION. 41 

51. Let us first consider the case in which both dividend and 
divisor are monomials. Take 

35a^62^2 tQ i3e divided by la^hc\ 
The operation may be indicated thus, 

—^iTi — j quotient, oa-^oc. 

id t/C 

Now, since the quotient must be such a quantity as multiplied 
by the divisor will produce the dividend, the co-efficient of the 
quotient multiplied by 7 must give 35 ; hence, it is 5. 

Again, the exponent of each letter in the quotient must be such 
that when added to the exponent of the same letter in the divisor, 
the sum will be the exponent of that letter in the dividend. 
Hence, the exponent) of a in the quotient is 3, the exponent of 
6 is 1, that of c is 1, and the required quotient is ^a^bc. 

Since we may reason in a similar manner upon any two 
monomials, we have for the division of monomij^Js the following 

RULE. 

I. Divide the co-efficient of the dividend hy the co-efficient of the 
divisor^ for a new co-efficient. 

II. Write after this co-efficient, all the letters of the dividend 
and give to each an exponent equal to the excess of its expo 
nent in the dividend over that iji the divisor. 

By this rule we find, 

48a%^c^d , ,, , 160a^Pcd^ r .-,. y 

T^TTz " = '^<^^<^(^ ; -^^T^ — = 5aWcd. 

I2ab^c SOa^d^ 

EXAMPLES. 

1. Divid 16.t;2 by Sx. Ans. 2x. 

2. Divide Xha'^xy'^ by 3ay. Ans. f>axy'^ 

3. Divide Mah'^x by 1262. ^,^5^ r^ahx. 

4. Divide —%Q,a^h'^c^ by \2a^c. An:. —Qa'^bc'^. 

5. Divide l^4.a%^c'd^ by —ZQa'^bHH. Ans. — 4a^b^cd^. 

6. Divide — 25Qa^c'^x^ by — IQa'^cx'^. Ans. I6abcx. 

7. Divide — SOOa^b^c^x^ by 30a*6Vrr. Aris. — lOahcx. 
8 Divide '-400a%^c^x^ by 25a^^c^x. Ans. - }Qbcx* 



42 ELEMENTS OF ALGEBRA. [CHAP. 11. 

52» It follows from the preceding rule that the exact division 
of monomials will be impossible : 

1st. When the cc-efficient of the dividend is not divisible by 
that of the divisor. 

2d. When, the exponent of the same letter is greater in the 
divisor than in the dividend. 

This last exception includes, as we shall presently see, the 
case in which tlie divisor has a letter which is not contained 
in the dividend. 

When either of these cases occurs, the quotient remains un- 
der the form of a monomial fraction ; that is, a monomial 
expression, necessarily containing the algebraic sign of division. 
Such expressions may frequently be reduced. 

Take, for example, -g-^- = -^. 

Here, an entire monomial cannot be obtained for a quotient; 
for, 12 is not divisible by 8, and moreover, the exponent of € 
is less in the dividend than in the divisor. But the expression 
can be reduced, by dividing the numerator and denominator by 
the factors 4, a^, 6, and c, which are common to both terms 
of the fraction. 

In general, to reduce a monomial fraction to its lowest terms: 

Suppress all the factors common to both numerator and dejiomir 
nator. 

From this rule we find, 

4:%aWcd^ _ 4a6^2 ^ Zlah^cH _ Zlb\ 

12aQ6 V _ 3«6 _7a26 _ 1 

' l6^^ - 4^' ^'''^* ~]AaW ~ 2ab ' 
In the last example, as all the factors of the dividend are 
found in the divisor, the numerator is reduced to 1 ; for, in fact, 
both terms of the fraction are divisible by the numerator. 

53. It often happens, that the exponents of certain letters, 

are the F*me in the dividei-d and divisor. 

24a362 
i^or example, - - - - _-, 



% 



CHAP. II.J DIVISION. 43 

is a case iu which the letter h is affected with the same expo- 
nent in the dividend and divisor : hence, it will divide out, and 
will not appear in the quotient. 

But if it is desirable to preserve th^ trace of this letter in 
the quotient, we niay apply to it the rule for exponents (Art, 
51), which gives 

The symbol 6°, indicates that the letter 6 enters times as 
ft factor in the quotient (Art. 16) ; or what is the same things 
that it does not enter it at all. Still, the notation shows that h 
was ill the dividend and divisor with the same exponent, and 
has disappeared by division. 

In like manner, ^ ^, ^ = ba%'^c^ = 55^. 

54i We will now show that the power of any quantity whose 
exponent is 0, is equal to 1. Let the quantity be represented 
by a, and let m denote any exponent whatever. 

Then, — = a"*~^ = a^. by the rule for division. 

But, — = 1, since the numerator and denominator are equal : 

hence, a° == 1, since each is equal to — • 

We observe again, that the symbol a° is only employed con- 
ventionally, to preserve in the calculation the trace of a letter 
which entered in the enunciation of a question, but which may- 
disappear by division. 

55. In the second place, if the dividend is a polynomial and 
the divisor is a monomial, we divide each term of the dividend 
hy the divisor^ and connect the quotients by their respective signs. 

EXAMPLES. 

Divide Qa^x^y^ — I2a^x^y^ -f Iha.'^x^y^ by Za'^x'^y^ 

Ans. 2x'^y* — 4aa;y* -f 5a*a;^y. 



44 ELEMENTS OF ALGEBRA. I CHAP. TL 

Divide 12ahj^ — 16aY + 20a6y* - 28a'^y^ by — 4a*y3. 

Ans. — Sy^ + 4a?/2 _ 5a2y ^ "7^3^ 

Divide i^a^c — 20acy2 -j- ^cd"^ "by — SaJc 

ah 

56. In the third place, when both dividend and divisor are 
polynomials. As an example, let it be required to divide 

2QaW + 10a* - 4Ba^b + 24a63 by 4a6 - ba^ + 362. 
In order that vre may follow the steps of the operation more 
easily, we will arrange the quantities with reference to the letter a. 

Dividend. Divisor. 

10a* — 48a36 + 26a262 + 24a53 | |— 5a2 + 4a6 + 352 

It follows from the definition of division and the rule for the 
multiplication of polynomials (Art. 45), that the dividend is the 
sum of the products arising from multiplying each term of 
the divisor by each term of the quotient sought. Hence 
if we could discover a term in the dividend which was derived, 
without reduction, from the multiplication of a term of the ^\wi 
sor by a term of the quotient, then, by dividing this term c/ 
the dividend by that term of the divisor, we should obtain one 
term of the required quotient. 

Now, from the third remark of Art. 46, the term 10a*, con 
taining the highest power of the letter a, is derived, without 
reduction from the two terms of the divisor and quotient, con- 
taining the highest power of the same letter. Hence, by dividing 
the term 10a* by the term — 5a2, we shall have one term of 
the required quotient. 

Dividend. Divisor. 



10a* — 48a36 + 26a262 + 24a53 
f 10a* — 8a36 — QaW 



'-5a2 4-4a6-{-362 



— 2a2 + 8a6 

— 40a36 + 32a262 -[_ 24a63 Quotient. 

— 40a35 j^ 32a262 _^ 24a63. 

Since the terms 10a* and — 5a2 are iffected with contrary 
signs, their quotient will have the sign — ; hence, 10a*, divided 
by — 5a2, gives — 2a2 for a eirm of the required quotient. 



CHAP. II.] ' DIVISION". 46 

After having wi'itten this term under the divisor, multiply each 
term of the divisor by it, and subtract the product, 

from the dividend. The remainder after the first operation is 
— 40a36 + 32a262 -f 24.aP. 

This result is composed of the products of each term of the 
divisor, by all the terms of the quotient which remain to be 
determined. We may then consider it as a new dividend, and 
reason upon it as upon the proposed dividend. We will there- 
fore divide the term — 40a^6, which contains the highest power 
of a, by the term — Sa^ of the divisor. 

This gives + 8«^ 

'or a new term of the quotient, which is written on the right 
of the first. Multiplying each term of the divisor by this term 
of the quotient, and writing the products underneath the second 
dividend, and making the subtraction, we find that nothing re- 
mains. Hence, 

— 2a2 + 8a6 or Sab — 2a^ 
is the required quotient, and if the divisor be multiplied by it, 
the product will be the given dividend. 

By considering the preceding reasoning, we see that, in each 
operation, we divide that term of the dividend which contains 
the highest power of one of the letters, by that term of the 
divisor containing the highest power of the same letter. Now, 
we avoid the trouble of looking out these terms by arranging 
both polynomials with reference to a certain letter (Art. 45), 
which is then called the leading letter. 

Since a similar course of reasoning may be had upon any two 
polynomials, we have for the division of polynomials the follo^^dng 

RULE. 

I. Arrange the dividend and divisor with reference to a certain 
letter, and then divide the first term on the left of the dividend by 
the first term on the left of the divisor, for the first term of the 
quotient ; multiply the divisor by this term and subtract the pro 
duct from the dividend. 



1 



46 ELEMENTS OF ALGEBRA. ' [CHAP. II 

II. Then divide the first term of the remainder hy the first term 
of the divisor, for the second term of the quotient ; multiply the 
divisor by this second term, and subtract the product from the 
result of the first operation. Continue the same operation until a 
remainder is found equal to 0, or till the first term of the remainder 
is not exactly divisible by the first term of the divisor. 

In the first case, (that is, when the remainder is 0,) the 
division is said to be exact. In the second case the exact diyi* 
sion cannot be performed, and the quotient is expressed by 
writing the entire part obtained, and after it the remainder with 
its proper sign, divided by the divisor. 

SECOND EXAMPLE. 



Divide 21x^y 


2 + 25:^:2^3 _|_ QSxy^ — 40y5 


- 56rr5 - 


ISx^y 


by 


52^2 __ Sx^ - 6xy. 














-'407/-\-68xy^-\-25xh/ 


4- 21a;3y2 


-I8x^y- 


■56x'\ 5y2. 


— 6xy- 


-8a;2 


— 40y^-\-48xy^+ 64a: V 




- 82/3 + 4a;y2 _ 


Zx^y-\- 


7a^ 


] st rem. 20xy^ - 


■ 39a:22/3 


+ 2\x^y'^ 










20xy^ - 


■ 24:Xhf 


- 32a:3y2 










2d rem. - — 


■ Ibx-^y^ + 53:r3y2 


- I8x^y 




- 


• 15a:2y3 _{_ lQy.3y2 


+ 24:X^y 








8d. rem. 


- 


35^ V • 


- 42x^y - 


-56x^ 








35a:3y2 


—4:2x*y - 


-56x^ 






Final remainder 


. 


-0. 







57. Remark. — In performing the division, it is not necessary 
to bring down all the terms of the dividend to form the first 
remainder, but they may be brought down in succession, as in 
the example. 

As it is important that beginners should render themselves 
familiar with algebraic operations, and acquire the habit of calcu- 
lating promptly, we will treat this last example in a different 
manner, at the same time, indicating the simplifications which 
should be introduced. These consist in subtracting each partial 
product from the dividend as soon as this product is formed. 



CHAP. II.] 



DIVISIOlft 



47 



1st rem. 20x7/^ — S9xh/-{-2lx^y^ 
2d rem. 



'+4a;y2_3a,.2y^72.3 



56a;S 



— 15a;2y3 _f_ 53^3^2 _ 18^4^ 

3d rem. .... S6x^y^ — ^2x*y 
Final remainder - . . 0. 

First, by dividing — 40?/^ by 5y^, we fibtain — Sy^ for the 
quotient. Multiplying 5?/^ by — Sy^, we have — 40y^, or, by 
changing the sign, + 40y^, which cancels the first term of the 
dividend. 

In like manner, — 6xy X — 8y^ gives + 4Sxy^, or, changing 
the sign, — 48.^?/*, which reduced with + QSxy"^, gives 2^xy^ for 
a remainder. Again, — ^x"^ x — 8^^ gives +, and changing the 
sign, — 642;2y3, -vv^hich reduced with 25x^y^, gives — ^9x^y^, 
Hence, the result of the first operation is 20xy* — S9x^y^, fol 
lowed by those terms of the dividend which have not been 
reduced with the products already obtained. For the second 
part of the operation, it is only necessary to bring down the 
next term of the dividend, to separate this new dividend fi^om 
the primitive by a line, and to operate upon this new dividend in 
the same manner as we operated upon the primitive, and so on. 



Divide 
4- 5 - 11a + 7a3. 

56a4 - 59a3 



THIRD EXAMPLE. 

95a - 73a2 + 56«* - 25 - 59a3 by -3a' 



73a2 + 95a — 25 1 



1st rem. 



35a3 + 15a2 + 55a — 25 



7a3-3a2- 11a + 5 



8a -5. 



2d remainder 



1 



0. 



GENERAL EXAMPLES. 



1. Divide 10a6 -f- 15ac by 5a. Ans. 2b + 3c 

2 Divide 30aa; — 54a; by Qx. Ans. 5a — 9. 

3. Divide lOx'^y — 15y2 — 5y by 5y. Ans. 2x^ — Sy — 1, 

4. Divide 12a + 3aa; — 18aa;2 by 3a. Ans. 4 + a; — 6x^. 



\S ELEMENTS OF ALGEBRA. [CHAP. U 

5. Divide Gax"^ + 9«^^ + ct^^^ by ax. Ans. Qx -{- 9a -{- ax. 

6. Divide a^ -f 2ax -{- x^ by a -{- x.- Ans. a -\- x, 

7. Divide a^ — Za^y ,-f- Zay^ — y^ by a — y. 

Ans. a^ — 2ay 4- y^. 

8. Divide 24a26 — 12fl3c&2 _ 6^^ by —^ah. 

Ans. —4:a + 2ahb + 1. 

9. Divide Qx^ — 96 by ^x — 6. Ans. 2x3 _}_ 4^2 ^ ^^ _^ jg. 

10. Divide - - a^ — haH + l^aH'^ — lOa^^jS + Saa;* — af« 
by a^ — 2ax + ^^. v4;i5. a^ — Sa^^r + Saa;^ — x^. 

11. Divide 48a;3 — 76a.'c2 _ 64^2^; + lOSa^ by 2a: — 3a. 

^'/is. 242;2 - 2a:c - 35a2. 

12. Divide y^ — ^y^x"^ + Sy^.-r* _ x^ by y^ — 3y2.r + 3yi:2 _ ^ 

^?is. y^ -\- Sy^x -{- 3ya;2 -f- x^. 

13. Divide ^ 64a456 - 25a2i8 ^y Sa'^P -\- dab''. 

Ans. Sa^3 — 5a6*. 

14. Divide Ga^ + 23a25 + 22a52 -|- 553 ^^ 3^2 _^ 4^5 _^ 52^ 

^'/i5. 2a -h 56. 

15. Divide Qax^ + Qax^y^ + 42a2.c2 "by arc + 5aa;. 

^?i5. x^ -{- xy^.-^7ax. 

16. Divide — 15a4 + 37a26c^ _ 29a^cf — 20b^d^ + 4Abcdf— %c'^n 
by 3a2 _ f)bd + r/. ^W5. — 5a2 4- 4:hd — 8c/. 

17. Divide x^ -\- x^y"^ + y* by a:2 — ary + y2. 

Ans. x^ -{- xy -}- y^. 

18. Divide x* — y* by a; — y. ^7i5. a:^ -f- a:2y 4- xy^ -{- y^. 

19. Divide 3a* — 8a252 4. 3a2,-2 + 554 __ 3J2c2 by a2 — 52, 

^715. 3a2 - 562 ^ 3^2, 

20. Divide 6x^ — 6x^y^ — Qx^ + 6a;3y2 + 15.r3y3 - 9.rV* 
■f 10a:2y5 ^ 15^5 by 3a;3 + 22;2y2 -|- 3y2. 

Ans. 2x3 _ 3^:2^2 _|_ 5y3 



CHAP. 11.3 DIVISION. 49 

REMARKS ON THE DIVISION OF POLYNOMIALS. 

58. The exact division of one polynomial by another is impossible : 

1st. When the first term of the arranged dividend or the first 
term of amj of the remainders^ is not exactly divisible by the first 
term of the arranged divisor. 

It may be added with respect to polynomials that we ciiu 
often discover by mere inspection that they are not divisible. 
When the polynomials contain two or more letters, observe the 
two terms of the dividend and divisor, which contain the highest 
powers of each of the letters. If these terms do not give an 
exact quotient, we may conclude that the exact division is iiu-- 
possible. 

Take, for example, 

12«3 - 5^26 + 7a62 _ 1153 | |4a2-{-8a6 + 35^. 

By considering only the letter a, the division would appear 
possible ; but regarding the letter 6, the exact division is impos- 
sible, since —llb^ is not divisible by 36^. 

2rf. When the divisor contains a letter which is not in the dividend. 

For, it is impossible that a third quantity, multiplied by 
one which contains a certain letter, should give a product inde- 
pendent of that letter. 

3c?, A monomial is never divisible by a polynomial. 

For, every polynomial multiplied by either a monomial or a 
polynomial gives a product containing at least two terms whicli 
are not susceptible of reduction. 

4:th. If the letter^ with reference to which the dividend is ar- 
ranged^ is not found in the divisor^ the divisor is said to be inde- 
yendent of that letter ; and in that case, the exact division is 
impossible, unless the divisor will divide separately the co-efficients 
of the different powers of the leading letter. 

For example, if the dividend were 

36a* 4- 96a2 -f 126, 
arranged with reference to the letter a, and the divisor 36, the 
divisor would be independent of the letter a; and it is evident 

4 



50 ELEMENTS OF ALGEBRA. [CHAP. IL 

that the exact division could not be performed unless the co- 
efficients of the different powers of a were exactly divisible by Zb, 
The exponents of the different powers of the leading letter 
ui the quotient would then be the same as in the dividend. 

EXAMPLES. 

1. Divide 18a%2_36a2a;3 — 12aa; by Qx. 

fAns. Sa^x — 6a^x^ — 2a. 

2. Divide 25a^b — SOa^ -\- 40ab by 6b. 

Ans. 5a* — 6a2 + 8», 
From the 3d remark of Art. 46, it appears that the teira of 
the dividend containing the highest power of the leading letter 
and the term containing the lowest power of the samfc letter 
are both derived, without reduction, from the multiplication of a 
term of the divisor by a term of the quotient. Therefore, nothing 
prevents our commencing the operation at the right Instead of 
the left, since it might be performed upon the terms containing 
the lowest power of the letter, with reference* to which the ar- 
rangement has been made. 

Lastly, so independent are the partial operations required by 
the process, that after having subtracted the product of the divi- 
sor by the first term found in the quotient, we could obtain 
another term of the quotient by arranging the remainder with 
reference to some other letter and then proceeding as before. 
If the same letter is preserved, it is only because there is no 
reason for changing it ; and because the polynomials are already 
arranged with reference to it. 

OF FACTORING* POLYNOMIALS. 

59. "When a polynomial is the" product of two or more factors, 
it i(? often desirable to resolve it into its component factors. 
This may oflen be done by inspection and by the aid of the 
fonculas of Art. 47. 

"When one factor is a monomial, the resolution may be effected 
hy writing the monomial for one factor, and the quotient arising 



CHAP. IL] DIVISION. 51 

from the division of the given polynomial oy this factor for tbo 
other factor. 

1. Take, for example, the polynomial 

ah -\- ac, 
In which, it is plain, that a is a factor of both terms : hence 
ah -\- ac = a {h -{- c), 

2. Take, for a second example, the polynomial 

ah^c + 5ah^ -\- ah'^c^. 

It is plain that a and h"^ are factors of all the terms : hence 

ah'^c + 5a53 ^ ^52^2 _ ^52 (^ + 56 -f c% 

3. Take the polynomial 25a* — SOa^S + 15a262 . it is evident 
that 5 and a^ are factors of each of the terms. We may, there- 
fore, put the polynomial under the form 

5^2 (^5^2 _ Qah + 362). 

4. Find the factors of Za% + ^cfic + I'^a^xy. 

Ans. 3a2 (6 -f 3c + Qxy\. 

5. Find the factors of ^a'^cx — ISaca;^ + 2ac^y — ZOa^c^x. 

Ans. 2ac (4aa; — dx"^ + c*?/ — 15a^c^x), 

6. Find the factors of 24:aWcx — S0a%^c^7j + SQa^^cd + 6a6<;. 

^7Z5. 6a6c {4ahx — ^d^h^c^y + Gae^'c^ + 1). 

By the aid of the formulas of Art. 48, polynomials having 
certain forms may be resolved into their binomial factors. 

1. Find the factors of a^ + 2ah + h'^. 

Ans. {a-\-b) X (a + b) 

2. 49a;* + 56x^y + IQx^y^ = {7x^ + 4.xy) {7x^ -f Axy), 

3. Find th-e factors of a^ — 2ah + h^. 

Ans. (a — 6) X (a — h^. 

4. 64a262c2 — AQabc'^d^ -\- 9c2# = {8ahc — Scd^) {Sahc — Zcd'^). 

5. Find the factors of a^ __ 52^ j^^s. {a + h) X (a — b), 
G 16a2c2 - 9(?* = (4ac + 3c?2) (4^^ - 3c?2). 



52 ELEMENTS OF ALGEBRA ICHAP. IL 

GENERAL EXAMPLES. 

1. Find the factors of the polynomial 6a% + Sa?b^ — 16a6' 

- 2ab. 

2. Find the factors of the polynomial 16abc^ — 'Sbc"^ -f 9a^^<fi 
~ 12db^c\ 

S. Find the factors of the polynomial 25a^c^ — ^Oa^c'^d 

— 5ac* — 60ac6. 

4. Find the factors of the polynomial 42a'^b^ — labcd + labd 

Ans. lab (6a& — cc? + d\ 

5. Find the factors of the polynomial n^ + 271^ + ^^ 
First, n^ + 2n^-{-n = n {n^ -{- 2n -\- I) 

= n{n +1) X {71+ 1) 
= n (/I 4- 1)2. 

6. Find the factors of the polynomial 5a^c + lOab'^c + ISaSc^. 

^n5. 5a6c (a + 25 + 3c). 
'7. Find the factors of the polynomial a'^x — rc^. 

Ans. X (a -{- x) {a — x), 

60. Among the different principles of algebraic division, there 
is one remarkable for its applications. It is enunciated thus : 

The difference of the same powers of any two quantities is Mcactlf 
. . . • 
divisible by the difference of the quantities. 

Let the quantities be represented by a and b ; and let m de 
note any positive whole number. Then, 

a"» _ 5« 
will express the difference between the same powers of a md ft, 
ssnd it is to be proved that a^ — b^ is exactly divisible bj a — 6. 
If we begin the iivision of 

a'" — S*" by a — 5, 
we have 



am — ^m I 



a — b 



fmr-l 



1st rem. a^ ^ — b'^ 

or, by factoring - - - 6(a"»~i — 6'""^). 



CHAP. II.] DIVISION. 53 

Dividing a*» by a the quotient is a*^"^, by the rule foi the 
exponents. The product of a — 5 by a^~'^ l/emg subtracted frona 
the dividend, the first remainder is a^~'^b — 6"*, which can be 
put under the form, 

b (a*"-! — i*"-!). 

Now, if the factor 

of the remainder, be divisible by a — 6, b times {a^*~^ — ^'^^)» 
must be divisible by a — b, and consequently a*" — b^ must 
also be divisible by a — b. Hence, 

If the difference of the same powers of two quantities is exactly 
divisible by the difference of the quantities^ then^ the difference oj 
the powers of a degree greater by 1 is also divisible by it. 

But by the rules for division, we know that a^ — b"^ is divis 
ible by a — 5 ; hence, from what has just been proved, a^ — b^ 
must be divisible by a — Z>, and from this result we conclude 
that a* — Z>* is divisible by a — 5 and so on indefinitely : hence 
the proposition is proved. 

61. To determine the form of the quotient. If we continue 
the operation for division, we shall find a^~% for the second 
term of the quotient, and a^~%'^ — b^ for the second remainder ; 
also, a^~W for the third term of the quotient, and a^~%^ — 5» 
for the third remainder; and so on to the m** term cf the quo 
tient, which will be 

and the m*^ remainder will be 

^m~m^m — Jwi qj. Jm — b"^ z= 0. 

Since the operation ceases when the remainder becomes 0, wo 
sha.l have m terms in the quotient, and the result may be -vvriV. 
ten thus : 
a"* — 6" 



a — 



— a»*-i -f a'»-26 + a'^-^^ + + a6»«-2 ■]- i*-'^ 



CHAPTER m. 



OP ALGEBRAIC FRACTIONS. 



62. An ALGEBRAIC FRACTION is an expression of one or more 
equal parts of 1. 

One of these equal p*rts is called the fractional unit. Thus, 

— is an algebraic fraction, and expresses that 1 has been divided 

into b equal parts and that a such parts are taken. 

The quantity a, written above the line, is called the numer- 
ator ; the quantity b, written below the line, the denominator; 
and both are called terms of the fraction. 

One of the equal parts, as — , is called the fractional unit; 

and generally, the reciprocal of the denominator is the frac- 
tional unit. 

The numerator always expresses the number of times that the 
fractional unit is taken ; for example, in the given fraction, the 

fractional unit -r- is taken a times. 



63. An entire quantity is one which does not contain any 
fractional terms ; thus, 

a^b + ex is an entire quantity. 
A mixed quantity is one which contains both entire and frao> 
tional terms ; thus, 

a'^b + — is a mixed quantity. 

Every entire quantity can be reduced to a fractional forra 
having a given fractional unit, by multiplying it by the denomi- 
nator of the fractional unit and then writing the product over the 
denominator ; thus, the quantity c may be reduced to a fractional 



CHAP. III.] ALGEBRAIC FRACTIONS. 55 

form with the fractional unit — , by multiplying c by 5 and 

be 
dividing the product by 6, which gives — . 

64» If the numerator is exactly divisible by the denominator, 
a fractional expression may be reduced to an entire one, by sim- 
ply performing the division indicated; if the numerator is not 
exactly divisible, the application of the rule for division will 
sometimes reduce the fractional to a ^lixed quantity. 

65* If the numerator a of the fraction — be multiplied by 

any quantity, q, the resulting fraction -~ will express q times 

a 

as many fractional units as are expressed by — ; hence : 

Multiplying the numerator of a fraction by any quantity is 
equivalent to multiplying the fraction by the same quantity. 

66. If the denominator be multiplied by any quantity, q^ the 
value of the fractional unit, will be diminished q times, and the 

resulting fraction — will express a quantity q times less than 

the given fraction ; hence : 

Multiplying the denominator of a fraction by any quantity ^ is 
equivalent to dividing the fraction by the same quantity, 

67» Since we may multiply and divide an expression by the 
same quantity without altering its value, it follows from Arts, 
65 and 66, that : 

Both numerator and denominator of a fraction may be multiplied 
by the same quantity, without changing the value of the fraction. 

In like manner it is evident that: 

Both numeratcr and denominator of a fraction may be divided 
by the same quantity without changing the value of the fraction, 

68. We shall now apply these principles in deducing rules 
for the transformation or reduction of fractions. 



56>. ELEMENTS OF ALGEBRA, [CHAP. UI 

I. A fractional is said to be in its simplest form when tlie numer- 
ator and denominator do not contain a common factor. Now, 
since both terms of a fraction may be divided by the same 
quantity without altering its value, we have for the reduction 
of a fraction to its simplest form the following 

RULE. 

Resolve both numerator, and denominator into their simple fao- 
tors (Art. 59) ; then, suppress all the factors common to both 
terms, and the fraction will be in its simplest form. 

Remark, — When the terms of the fraction cannot be resolved 
into their simple factors by the aid of the rules already given, 
resort must be had to the method of thp. greatest common divi 
£or, yet to be explained. 

EXAMPLES. 

1 T-» T 1 r> • ^^^ "4" 6ac . . -, „ 

1. Keduce the fraction ^ — , , ^^ tf fs simplest form. 
Sad + 12a ^ 

We see, by inspection, that 3 and a f»^' focto'^i of the nu- 
merator, hence, 

Sab + Qac = Sa {b -\- 2€\ 

We also see, that 3 and a are factors v/" the d.eiOTaiimi'<*t 

hence. 

Sad + 12a = Sa{d-\- 4). 

3a5 + 6ac _ 3a (5 + 2c) _ 6 -f ?-r 
^^^®' 3a^ + 12a ~ 3a {d + 4) "" d -\- 4" 



2. Reduce r-^ — . ^ , to its simplest form. 
9ab -\- Sad 



Ans. 



256c "h obf 
3. Reduce ^^„ , .. ;, to its simplest form. 
300^ H- 156 ^ 



2ab f g 
n -hi 



Ans ^'±J. 

_, , 54a6c . . -, ^ 

4. Keduce — -r to its simplest form. 

Ans. ,; — -— 
aa -\- a 



CHAP. III.] 



ALGEBRAIC FRACTIONS. 



57 



5. Reduce 



(i. Reduce 



SQa^ + I2abf 
84a62 



to its simplest form. 



Am. 



76 ' 



12acd — 4cJ2 



to its simplest form. 



Ans 



3a 



3/4- c* 



^ ^ , 18a2c2 — 3ac/ . . -, ^ 

7. Reduce ^^ ^ -~ to its simplest form. 

27ac^ — 6ac-^ 

6ac — / 

' 9c — 2c2" 

II. From what was shown in Art. 63, it follows that we may- 
reduce the entire part of a mixed quantity to a fractional form 
with the same fractional unit as the fractional part, by multiply- 
ing and dividing it by the denominator of the fractional part. 
The two parts having then the same fractional unit, may be 
reduced by adding, their numerators and writing the sum obtained 
over the common denominator. 

Hence, to reduce a mixed quantity to a fractional form, we 
have the 

RULE. 

Multiply the entire part by the denominator of the fracticm 
then add the product to the numerator and write the sum over the 
denominator of the fractional part. 



EXAMPLES. 



1. Reduce x — - 



to the form of a fraction. 



Here, 



a^ ~x^ _ x^ — (g^ — a;2) ^ ^x^ _ a» 

X X X ' 



ax ~f~ X 
a. Reduce x to the form of a fraction. 



2a 



Ans. 



ax — x^ 



68 



S. Reduce 



i. Reduce 



ELEMENTS OF ALGEBRA. 

2x-7 



[CHAP IIL 



Sx 



to the form of a fraction. 

17a; -7 



Ans. 



Sx 



5. Reduce 1 + 2a; 



6. Reduce Zx — I 



bx 



to the form of a fraction. 

2a - a; -f- 1 

Ans. . 

a 

to the form of a fraction. 

10a;2 + 4a. 4. 3 



-47Z5. 



5a; 



x -\- a 
2 



3a 



to the form of a fraction. 
9aa; — 4a — 7a; + 2 



Ans. 



3a — 2 



Remark. — "We shall hereafter treat mixed quantities as though 
they were fractional, supposing them to have been reduced to a 
fractional form by the preceding rule. 

III. — From Art. 64, we deduce the following rule for reduciijg 
a fractional to an entire or mixed quantity. 

RULE. 

f 
Divide the numerator by the denominator, and continue the oper 

ation so long as the first term of the remainder is divisible by the 

first term of the divisor : then the entire part^ of the quotient found^ 

added to the quotient of the remainder by the divisor, will be the 

mixed quantity required. 

If the remainder is 0, the division is exact, and the quotient 
is an entire quantity, equivalent to the given fractional expres- 
sion. 



EXAMPLES. 



1. Reduce 



ax 4- «' 



to a mixed quantity. 



Ans. =i a -\ • 



CHAP, III.J ALGEBRAIC FKACTIONS. 69 

2. Keduce to an entiie or mixed quantity. 

Ans. a — a;. 



3. Keduce to a mixed quantity. 



4. Eeduce to an entire quantity. 



/Jj3 y3 

5. Reduce — to an entire quantity. 



2a» 
Ans. a 7-. 



A71S. a -\- X, 



x — y 



Ans. x"^ -\- xy -\- y^. 



0. Keduce to a mixed quantity. 

DX 

Ans. 2a; — 1 H- -— . 
bx 

rV. To reduce fractions having different denominators to equiv 
alent fractions having a common denominator. 

Let —, — and — , be any three fractions whatever. 
a J 

It is evident that both terms of the first fraction may be mul 

tiplied by df giving r—^ and that this operation does not 

change the value of the fraction (Art. 67). 

In like manner both terms of the second fraction may be 

hcf 
multiplied by bf^ giving j^ ; also, both terms of the fraction 

— may be multiplied by hd^ giving ■— .. 

-.- , , , /. . o.df hcf , hde 

If now we examine the three fractions ^-j^, 7^ and 7-7^ 

bdj bdj bdf 

we see that they have a common denominator, bdf^ and that 
each numerator has been obtained by multiplying the numerator 
of the correspcnding fraction by the product of all the denom- 
inators except its own. Since we may reason in a similar 
manner upon any fractions whatever, we have the following 



60 ELEMENTS OF ALGEBRA. [CHAP. IIL 

RULK 

Multiply each numerator into the p^oauct of all the denomina- 
tors except its own^ for new numerators^ and all the denominators 
together for a common denominator. 

EXAMPLES. 

1. Eeduce — and — to equivalent fractions having a com 

mon denominator. 

a X c z=zac) , 

TO r the new numerators. 
h xh = h^) 

and - h X c =zhc the common denominator. 

2. Reduce — ^^<i to equivalent fractions having i, com 

ac ^ ah ^ W- 

mon denommator. Ans. -r— and . 

be be 

Sx 25 

3. Reduce •^, — and c?, to equivalent fractions having a 

ica tic "^ 

T . . 9cx 4ab ^ 6acd 

common denommator. Ans. — — , --^ — and — — ■. 

bac bac Qac 

4. Reduce — , -— and a -\ , to equivalent fractions hav- 

Tc o a 

, 9a Sax ^ 12a'^ + 24tx 

ms a common denommator. A7is. -— -, -— - and — . 

^ ♦ 12a' 12a 12a 

5. Reduce -— -, — - and , to equivalent fractions hay 

2 3 a-{-x ^ j 

ing a common denominator. 

3a + 3a; 2a3 + 2a'^a; 6a^ ^ 6x^ 

^'''' 6M=^' ea + 6x ^"""^ 6a 4- 6a; ' 

6. Reduce -, , and — , to equivalent fractions hav 

a — b ax c 

mg a common denominator. 

. a^cx ac^ — ahc — hc^ -\- ch"^ ^ a^hx — a}p-x 

Ans. — r -— and — — , 

a'^cx — abcx a^cx — abcx a?cx - abcx 



CHAP. III.J ALGEBKAIC FKACTIONS. 61 

V. To add fractions together. 

Quantities cannot be added together unless they have the 
same unit. Hence, the fractions must first be reduced to equiv- 
alent ones having the same fractional unit; then the sum of 
the numerators will designate the number of times this unit 
Is to be taken. We have, therefore, for the addition of frao- 
tions the following 

RULE. 

Reduce the fractions^ if necessary^ to a common denom.inator : 
then add the numerators together and place their sum over the 
common denominator. 

EXAMPLES. 

1. Find the sum of -— , —=- and — r. 

h d f 

Here, - a X d xf = adf\ 

c X b Xf = cbf I the new numerators. 

e X b X d = ebd) 
And - b X d xf = bdf the common denominator. 
T-T adf . cbf , ebd adf -\- cbf -{• ebd , 

2. To «-^ add h + ^. Ans. a + i + ^^^i^^. 

be be 

XXX X 

3. Add — , — and — together. Ans. x -\- —-. 
2 , 4a; . ., , 19a; -14 



4. Add — -— and — together. 



Ans. 



21 



5. Add X -\ ^— - to Sx H — . Ans. 4a; -{ — . 

o 4 12 

f1* Q* I ft 

6. It is required to add 4a;, — — and — ^r — together. 

lia liX 

, ^ , 5a;3 -\- ax -\- a^ 

Ans. 4a; -\ . 

iZcix 

7. It IS required to add — > -— and — - — together. 

3 4 5 

49x + 12 



62 ELEMENTS OF ALGEBRA. [CHAP. ITT. 

1x X 

8. It is rec^uired to add 4a;, — and 2 4^ together. 

44a; + 90 

Ans. 4x H . 

45 

9. It is required to add 3a; + — and x — — together. 

A o , 23a; 
Ans. Sx + — — . 
45 

a ~~" X c d 

10. What is the sum of •,, — — r and 



Ans. 



a — b a -\- b a -\- x' 

a^ — ax"^ + o?b — bx"^ + a^c + acx — abc — bcx + a^d — b'^d 



a^ — b^a + tt-^a; — b'^x 
_ a^ + a2 (5 _^ c -{-d) — a {x^ — ex + be) — b {x'^ -\- ex -\- bd) 
a^ + a^x — ab'^ — b'^x 

VI. To subtract one fraction from another. 

Reduce the fractional quantities to equivalent ones, having the 
same fractional unit ; the difference of their numerators will 
express how many times this unit is taken in one fraction naore 
than in the other. Hence the following 

RULE. 

I. Reduc€ the fractions to a common denominator, 

II. Suhtraet the numerator of the subtrahend from the numer- 
ator of the minuend y and place the difference over the common 
denominator. 

EXAMPLES. 

• -T-. ^ — ^ 1 . 2a — 4a; 

1. From - - - subtract — . 

Zo 6c 

^^ (x— a) X 3c = 3ca; — 3ae ] , 

Here, .\ . \ ^, . , ^- V the numerators. 
' (2a — 4a;) X 2b = 4ab — 86a; ) 

And, 26 X Sc — 6bc the common denominatoi 

3ca; — Sac 4ab — Sbx Sex — Sac — 4a6 + Sbx 



Hence, 



66c 66c 66c 



^ T. 12a; .. 3a; ^ 39« 

2. From - - -=— subtract — . ATiS, -^, 



CHAP. III.] 


ALGEBRAIC FRACTIONS. 




ea" 


8. From - 


- 5y 


3y 
subtract ~. 

o 


Ans. 


37y 
8 • 


4. From - 


Sx 

- T 


subtract — . 

y 


Ans. 


13rr 
63* 


5, From - 


x + a 
b 


subtract -7-. Ans, 
d 


dx-\-ad 

bd 


-6c 

. 


6. From - 


Sx+ a 
5b 


subtract — - — . 











Ans. 


- lObx - 
406 


-356 


T. From - 


.3.-,| 








c 








Ans. 2x + 


cx-\-bx ■ 


-ab 



be 

VII. To multiply one fractional quantity by another. 
Let - represent any fraction, and - any other fraction; and 
let it be required to find their product. 

If, in the first place, we multiply - by c, the product will 

CLC 

be — , obtained by multiplying the numerator by c, (Art. 65) ; 

but this product is d times too great, since we multiplied 

- by a quantity d times too great. Hence, to obtain the true 

product we must divide by c?, which is effected (Art. QQ) by 

multiplying the denominator by d. We have then. 

a c ac . 

T X -^ = ri'i hence 

6 d bd 

* 

RULE. 

I. Cancel all factors common to the numerator and denomi^ 
nator. 

II. Multiply the numerators together for the numerator of the 
'product, and the denomhiators together for the denominator of the 
product. 



64 ELEMENTS OF ALGEBRA. [CHAP. IlL 



EXAMPLES. 



I. Multiply a-\ by — . 



Pirst, - - . - aH = 



-r-r a} -\-hx c a?-c + bcx 

Hence, - . x — r = ; 

a a ad 

2. Eequired the product of — and — . 

4/ 



2x Sx^ 

S. Required the product of — and -— . 



4. Find the continued product of — , and 

^ a c 



5. It is required to find the product of 6 H and — . 

ab + bx 



Ans. 


9ax 
2b' 


Ans, 


5a' 


Sac 




2b' 




Ans. 


9arr 



Ans. 



X 



x^ 52 a;2 _|_ 52 

6. Required the product of — r and — . 



a;* — 6* 
Ans. 



bH + b<^' 



X -\- \ X — 1 

7. Required the product of x -\ and , . 

ax^ — ax -\- x"^ — 1 

Ans. ^—. — 7 -. 

a^ -f- «o 

ftOR CI ■- - ^^ 

8. Required the product of a H and — T~2' 

a2 (a + rr) 



CHAP. III.] ALGEBRAIC FRACTIONS. 66 



VIII. To divide one fraction by another, 

Let -r- represent the first, and - 
< 

he division may be indicated thus. 



o, c 

Let — represent the first, and — the second fraction; then 



(i) 



[f now we multiply both numerator and denominator of tliis 
complex fraction by — , which will not change the value of the 

fraction (Art. 67), the new numerator will be 7—, and the new 

be 

cd 
denominator -7-, which is equal to 1. 

dc ^ 



a c _\h I \ bcj ad 

' h ' d ~ ( c\ ~~ \ ^ he' 



(i) © 



This last result we see might have been obtained by inverting 
the terms of the, divisor and multiplying the dividend by tlie 
resulting fraction. Hence, for the division of fractions, we have 
the following 



RULE. 



Invert the terms of the divisor and multiply the dividend by i/te 
resulting fraction. 



EXAMPLES. 



i. iDivide - - - a — •— • by — . 

b _2ac — b 

^ ~ 2^ ~ 2c 

rx ^ / 2ac — b q 2ac(j — bg 

Hence, a - — -^ ^ = — X ^ = — ITT^- 

2c g 2c f 2c/ 

2. Let —- be divided by — -. Ans, -— -. 

5 "^ 13 00 



66 


ELEMENTS OF ALGEBRA. 


[CHAP. III. 


^ 3. Let 


4^2 

— - be divided by bx. 




A 4^ 


4. Let 


— - — be divided by — . 

v) o 




A ^+1 

^^^- 4x ' 


5. Let 


be divided by -— . 

X — I "^ 2 




A 2 

X — 1 


G. Let 


y be divided by -^. 




hbx 
Ans. ^^. 


7. Lot 


x — b ....... ^cx 

^ , be divided by — — . 
Scd ^ 4:d 




A ^-* 


8. Let 


/^.4 54 

— — — — r^ be divided by 

x^ — 2bx + b" 


x^ 

X 


+ bx 

-b ' 

Ans. X i . 

X 








9. Divide — by ^. Ans. 

\ —X -^ 1 — X^ 


ax(l -\- x) —X — 1 
a 


10. Divide "+] by ] + \. 

a— 1 "^ \ —a?- 




Ans. — (1 4 a). 


69. If 


we have a fraction of the form 
a 




wc may 


observe that 






— a , a 

- = — c, also -- = — c and 

— 


— 


- — c ; that is, 



The sign of the quotient will be changed by changing the sign 
■either of the numerator or denominator^ but will not be affected by 
■changing the signs of both the terms. 

70 • We Avill add two propositions on the subject of fractions. 

I. If the same number be added to each of the terms of a proper 
fraction^ the fraction resulting from these additions will be greatej 
than the first ; but if it be added to the terms of an impropei 
fraction^ the resulting fraction will be less than the first. 

Let the fraction be expressed by — . 

Let m represent the number to be added to each term : the» 

Ihe new fraction will be, 7— — . 

-\- m 



CHAP III.] ALGEBRAIC FRACTIONS. 67 

In order to compare the two fractions, they must h^ reduced 

to the same denominator, which gives for 

a ab -\- am 

the nrst traction, -— = ^^ , , 

b b^ -j- bm 

a-^m ab-\-bm 

and for the new fraction, ■; — ; — = r——~ — . 

b -\- m b^ -\- bm 

Now, the denominators being the same, that fraction will he 

the greater which has the greater numerator. But the two 

numerators have a common part a6, and the part bm of the 

second is greater than the part am of the first, when 6 > « : 

hence 

ab -\- bm ^ ab -{- am ; 
that is, when the fraction is proper, the second fraction is greater 
than the first. 

If the given fraction is improper, that is, if a > 6, it is plain 
that the numerator of the second fraction will be less than that 
of the first, since bm would then be less than am. 

II. If the same number be subtracted from each term of a proper 
fraction^ the value of the fraction will be diminished; but if it he 
subtracted from the terms of an improper fraction^ the value of the 
fraction luill be increased. 

Let the fraction be expressed by — , and denote the number 

to be subtracted by m. 

Then, will denote the new fraction. 

b — m 

By reducing to the same denominator, we have, 

a ab — am 



and 



a — m ab — bm 



b — m 6^ — bm 
Now, if we suppose a < b. then am<^bm\ and if ttwi < hm,^ 

then will 

ab — am "^ ab — bm\ 

that is, the new fraction will be less than the first. 

If a > 6, - that is, if the fraction is improper, then 

am > bm^ and ah — am <^ab — bm^ 

that is, the new fraction will be greater than the first. 



68 ELEMENTS OF ALGEBRA. LCHAP. Ill 



GENERAL EXAMPLES. 



1 4- aj2 1 _ a;2 2 (1 + a:*) 

2. Add =-^; — to . Ans, 



\ \-x \-x 1 - aj2 

„ _ a 4- 6 , a — h . 4a6 

3. From — ~- take — — r- -^^5- 



a-6 ' a + 6* • a2-52* 

, ^ 1 + a;2 ^ , 1 - rc2 4a;2 

4. From -. take t-j— 5. Ans. -. 

I — x^ I A- ^ 1 — «* 

ar2-llrr4-28 

-dwS. :: . 



^ ,r , . T X* — h* . x^ -^ bx . ft . TO 

6. Multiply ^^jjqrp l>y ^ri- ^"'- ^ + ^''- 

7. Divide «_+5 + ^ by ^±f-^. 

a — ic a -\- X a — x a -^ x 

a^-\-x^ 



Ans. 



2ax 



8. Divide 1 -\ -— - oy 1 — r-. Ans. n. 



EXAMPLES INDICiTLNG USEFUL FORMS OF REDUCTION. 



Tx'^ dx"^ "*"^3 ~ bdfx^ "^ b^^ "*' 6^ 

ac?/a;2 + 6c/a; -f- bde 

a c e g __ adfhx^ bcfhx^ bedhx^ bdfgy^ 

Tx "*" dx'^^f^~kx^ ~ bdfkx^^ "^ bdfhx^^~bdfh^^^Mfh3?^ 
adfhx^ + bcfhx"^ — beihx — hdfg 
"" ~ bdfh^ ' 



lOHAP. III. EXAMPLES IN FRACTIONS. 69 


l-\-x^ l-X^ 
1 - a:2 ' 1 + a:2 


(1+^2)2 (l_:r2)2 

(1 4- x^Y 4- (1 - rr2)2 
- (1 - X') (1 + 0:2) 

2(1 4- x^) 
" 1-x^' 


^' 1 + a: ' 1-x 


1—X 1 + X 


-(1 + ^)(1-^) 

2 
- 1-x^' 


\ 


a + 6 a — b 
' a — b a -{- b 


(a + 5)2 -(a -5)2 
- ^a-\-b){a-b) 
4ab 


1+ a;2 1 _ a;2 
l-x^ 1 + a;2 


(1 + i^:2)2 (1 _ ^2)2 
-(1_^2)(14.^2) (1_^2)(14.^,2) 
(1 4- ^2)2 _ (1 _ X^Y 

- (l_a;2)(l+a;2) 

4a;2 
" 1 - a:^ * 


1 _ a;2 • 1 + a;2 


l-{-x'^ ^ l-\-x^ (1 + x^Y 
" l-x^ '^ l-x^ ~ (1 - a:2)2- 


6. -^-^^^.i 
a;2 - 2bx 4 62 • 


v^+hx x^ — b^ .. ^-6 
a;_ 5 - a;2 _ 262? + 52 ^^2 - 6a; 
_ (a:* -^ 64) (^ _j) 




"" (x^ - 26a; + 62) (a;2 + 6a:) 
(a;2 _ 62) (a:2 4- 62) (a; — 6) 

- {x- 6)2 a; (a; 4- 6) 

(a; 4- 6) (a: — 6) (a;2 4- 62) (x — b) 




~ x{x - 6) {x ^ 6) (a; 4- *) 
ar» + 62 



70 KLEMENTS OF ALGEBRA. LCHAP. UL 

Of the Symbols 0, go and — . 

71. The symbol is called zero, which signifies in ordinary 

language, nothing. In Algebra, it signifies no quantity : it is 

also used to expres a quantity less than any assignable quantity. 

Tlie symbol oo- is called the symbol for infinity ; that is, it is 

used to represent a quantity greater than any assignable quantity. 

If vre take the fraction — , and suppose, whilst the value of 

a remains the same, that the value of b becomes greater and 
greater, it is evident that the value of the fraction will become 
less and less. When the value of b becomes very great, the 
lvalue of the fraction becomes very small; and finally, when b 
becomes greater than any assignable quantity, or infinite, the 
value of the fraction becomes less than any assignable quantity, 
or zero. 

Hence, we say, that a finite quantity divided by infinity is 
equal to zero. 

We may therefore regard — , and 0, as equivalent symbols. 

If in the same fraction — , we suppose, whilst the value of a 

remains the same, that the value of b becomes less and less, it 
is plain that the value of the fraction becomes greater and 
greater; and finally, when b becomes less than any assignable 
quantity, or zero^ the value of the fraction becomes greater than 
any assignable quantity, or infinite. 

Hence, we say, that a finite quantity divided by zero is equal 
to infinity. 

We may then regard — and oo as equivalent symbols : Zerc 
and infinity are reciprocals of each other. 

The expression — is a symbol of indetermination ; that is, ifc 

is employed to designate a quantity which admits of an infinite 
number of values. The origin of the symbol will be explained 
hi the next chapter. 



(?HAP. III. J ALGEBRAIC FRACTIONS. 71 

It should be observed, however, that the expression — is not 

always a symbol of indetermination^ but frequently arises from 
the existence of a common factor^ in both terms of a fraction, 
which factor becomes zero, in consequence of a particular hypo- 
thesis. 

1. Let us consider the value of x m the expression 



a2 _ 52' 

If, in this formula, a is made equal to 5, there results 



" = ¥• 

But, - - - a? — h^^{a — b) (a2 + «§ + ^2) 
and - . a2 — 52 = (a - h) {a, + 5), 
hence, we have, 

_ (g - h) (a2 J^^ab + 62) 
^ - {a - 6) (a + b) • 

Now, if we suppress the common factor a — 6, and then sup 
pose a = b, we shall have 

3a 

" = -¥■ 

2. Let us suppose that, in another example, we have 
_ a2 - &2 
""-(a-by 
If we suppose a = b, we have 



If, however, we suppress the factor common to the numerator 
and denominator, in the value of x, we have, 
_ {a-\-b){a — b) a -\-b 
~ {a ~b){a — b)~ a —b' 
If now we make az=.b^ the value of x becomes 

26 



If now we make a=zh, the value of x becomes 



72 ELEMENTS OF ALGEBKA. [CHAP. Ill, 

3. Let us suppose in another example, 

"^ - a3 - 63 ' 

ill which the value of x becomes — when we make a =. 6. 

If we strike out the common factor a — 6, we shall find 

a — h 
^~a2-f ab + 62- 

e 

Therefore, before pronouncing upon the nature of the expres 

sion — , it is necessary to ascertain whether it does not arise 

from the existence of a common factor in both numerator and 
denominator, which becomes under a particular hypothesis. 
If it does not arise from the existence of such a factor, we 
conclude that the expression is indeterminate. If it does arise 
from the existence of such a factor, strike it out, and then make 
tJie particular supposition. 

If A and B represent finite quantities, the resulting value of 
the expression will assume one of the three forms; that iS; 
A A 

it will be either finite^ infinite^ or zero. 

This remark is of much use in the discussion of problems. 



CHAPTER IV. 

EQUATIONS OP THE FIRST DEGREE INVOLVING BUT ONE UNKNOWN QUANTITY. 

72. An Equation is the algebraic expression of equality bts 
tween two quantities. 

Thus, a; = a + ^, 

is an equation, and expresses that the quantity denoted by x i« 
equal to the sum of the quantities represented by a and 6, 

Every equation is composed of two parts, connected by the 
sign of equality. The part on the left of this sign is called the 
first member^ that on the right the second member. The second 
member of an equation is often 0. 

73. An equation may contain one unknown quantity only, or 
it may contain more than one. Equations are also classified 
according to their degrees. The degrees are indicated by ths 
exponents of the unknown quantities which enter them. 

In equations involving but one unknown quantity^ the degree is 
denoted by the exponent of the highest power of that quantity in 
any term. 

In equations involving more than one unknown quantity^ tht 
degree is denoted by the greatest sum of the exponents of the unknown 
quantities in any term. 

For example: 

~[ ~ , , ^ , ^ \ are equations of the first degree. 
ax + ooy + C2 + oa = ) 

aa;2 + 26^ + c = ) 

„ . , . „ , /^ r are equations of the second degree. 
ax^ + bxy -{- cy^ -^ d = d ) ^ 

^ ~ , „ [• are equations of the third degree, 

4:axy^ — 2cy3 -|- abxy = 3 ) ^ 

and so on. 



74 ELEMENTS OF ALGEBRA. [CHAP. IV 

74. Equations are likewise distinguished as numerical equations 
and literal equations. The first are those which contain numbers 
only, with the exception of the unknown quantity, which is 
always denoted by a letter. Thus, 

A:X — Z = 2x-{- 5, Zx^ — x^ 8, 
are numerical equations. 

A literal equation is one in which a part, or all of the known 
quantities, are represented by letters. Thus, 

hx"^ -\- ax — 3j; = 5, and ex + dz^ = c + /, 
are literal equations. 

75i An identical equation is an equation in which one member 
is repeated in the other, or in which one member is the result of 
certain operations indicated in the other. In either case, the 
equation is true for every possible value of the unknown quan- 
tities which enter it. Thus, 

a;2 _ y2 
.cx -{- h =z ax -\- 5, [x 4- clY = x'^-{- 2ax + ot^ = a; — y, 

X+7/ 

are identical equations. 

76. From the nature of an equation, we perceive that it must 
possess the three following properties: 

1st. The tsvo members must be composed of quantities of the 
same kind. 

2d. The two members must be equal to each other. 

3d. The essential sign of the two members must be the same. 

76.* An axiom is a self-evident proposition. We may here 
enumerate the following, which are employed in the transforma- 
tion and solution of equations : 

1. If equal quantities be added to both members of an equation, 
the equality of the members will not be destroyed. 

2. If equal quantities be subtracted from both members of an 
equation, the equality will not be destroyed. 

3. If both members of an equation be multiplied by equal 
quantities, the products will be equal. 

4. If both members of an equation be divided by equal quan 
titles, the quotients will be equal. 

5. Like powers of the tM'O members of an equation are, equal 

6. Like roots of the two members of an eqaatiou are eauaL 



CHAP. IV.] EQUATIONS OF THE FIEST DEGREE. 75 

Soluticm of Equations of the First Degree, 

77. The solution of an equation is the operation of finding a 
value for the unknown quantity such, that when substituted for 
tne unknown quantity in the equation, it will satisfy it ; that iS;, 
make the two members equal. This value is called a root of 
the equation. 

In solving an equation, we make use of certain transformations. 

A transformation of an equation is an operation by which we 

change itf form without destroying the equality of its members. 

First Transformation. 

78. The object of the first transformation is, to reduce an 
equation^ some of whose terms are fractional^ to one in V)hich all 
of the terms shall he entire. 

Take the equation, 

Eirst, reduce all the fractions to the same denominator, by the 
known rule ; the equation then becomes 

72 ~ "72 "*" "72 ~ * 
If now, both members of this equation be multiplied by 72, 
the equality of the members will be preserved (axiom 3), and 
the common denominator will disappear ; and we shall have 

48a; — 54a; + 12a; nr 792 ; or by dividing 
both members by 6, 8a; — 9a; + 2a; = 132. 

The last equation could have been found in another manner 
by employing the least common multiple of the denominators. 

The common multiple of two or more numbers is any num- 
ber which each will divide without a remainder; and the least 
common multiple, is the least number which can be so divided. 

The least common multiple of small numbers - can be found 
by inspection. Thus, 24 is the least common multiple of 4, 6 
and 8; and 12 is the least common multiple of 3, 4 and 6. 



76 ELEMENTS OF ALGEBRA. [CHAP. IT. 

Take the last equation, 

T-T^ + T^i'- 

We see that 12 is the least common multiple of the de- 
nominators, and if we multiply each term of the equation by 
12, reducing at the same time to entire terms, we obtain 

8a; — 9a; + 2x = 132, 
the same equation as before found. • 

Hence, to transform an equation involving fractional terms to 
one involving only entire terms, we have the following 

RULE. 

Form the least common multiple of all the denominators^ and 
then multiply both members of the equation by it, reducing fractional 
to entire terms. 

This operation is called clearing of fractions. / 



EXAMPLES. 
X X 

1, Reduce -^ + -7 3 = 20, to an equation involving only 

eitire terms. 

We see, at once, that the least common multiple is 20, by 
which each term of the equation is to be multiplied. 

Now, — X 20 = a; X — = 4a;, 

o o 

X 20 

and --x20=:a;x-7- = 5a;: 

4 4 

that is, we reduce the fractional to entire terms, by multiplytny 

the numerator by the quotient of the common multiple divided by 

the denominator, and omitting the denominators. - 

Hence, the transformed equation is 

4a; + 5a; — 60 = 400. 

XX 

2. Reduce — + -= 4 = 3 to an equation involving only 

entire terms. Ans. Ix-^-Ktx — 1 40 = 105. 



CliAF, ly.J EQUATIONS OF THE FIRST DEGREE. 77 

CL C 

3. Reduce — — +/=y to an equation involving only 

entire terms, Ans. ad—hc-\- bdf= hdg. 

4. Reduce the equation 

ax 2c'^x , , 4'bc^x 5a^ 2c^ 

J- -y^a = — -—-\ 36 

(Jib ' a^ b^ a 

to one involving only entire terms. 

Ans. a'^hx — ^a^hc^x -|- 4a^62 _ 453^2^ _ 5^6 _^ ^aWc^ — Za^¥. 

Secoiid Transformation. 

79. The object of the second transformation is to changf. 
any term from one member of an equation to the other. 

Let us take the equation 

ax ■\- h -j:x d — ex. 
[f we add ex to both member:?, the equality will not be de- 
stroyed (axiom 1), and we shall liave 

ax -\- ex -\- b =z d — ex -\- ex '^ 
or by reducing, ax -\- ex -\- b ■= d. 

Again, if we subtract b from both members, the equality 
will not be destroyed (axiom 2), and we shall have, after 
r<iduction, 

ax -{■ ex = d — b. 
Since we may perform similar operations on any other equation, 
we have, for the change or transposition of terms, the following 

RULE. 

Any term of an equation may be transposed from one member 
to the other by changing its sign. 

80. We will now appi/ the preceding principles to the solu- 
tion of equations of the first degree. 

For this purpose let us assume the equation 
a + b ^ _ ^ _ ^^ _ a + d 
c ~ a ' 

Clearing of fractions, we have, 

a{a-\-h)x — acd = obex — c (a -f- d). 



78 ELEMENTS OF ALGEBEA, [CH^P. IV. 

If, now, we perform the operations indicated in both members, 
we shall obtain the equation 

<j?x + oh^ — CLcd = obex — ca — cd. 
Transposing all the terms containing x, to the first member, 
and all the known terms to the second member, we shall have, 
a^x + dhx — obex = acd — (lc — cd. 
Factoring the first member, we obtain 

(a^ -^ ah — ahc) x = acd — ac — cd : 
If we divide both members of this equation by the co- 
efficient of X, we shall have 

acd — ac — cd 
a^ •\- ab — ahc 
Any other equation of the first degree may be solved in a 
similar manner : 

Hence, in order to solve any equation of the fii'st degree, 
we have the following 

RULE. 

I. Clear the equation of fractions^ and perform in both members 
all the algebraic operations indicated. 

n. Transpose all the terms containing the unknown quantity/ to 
the first member, and all the known terms to the second member., 
and reduce both members to their simplest form. 

m. Resolve the first member into two factors, one of which shall 
be the unknown quantity ; the other one will be the algebraic sum 
of its several co-efficients. 

rV. Divide both members by the co-efficient of the unknown quan- 
tity ; the second member of the resulting equation will be the re- 
(piired value of the unknown quantity. 

1. Take the numerical example 

5a; __ 4rc _ 1 IZx 

12 T ~ "'"F 6~* 
Clearing of firactions 

10a; — 32a; — 312 = 21 — 52a;; 



CHAP. IV.J EQUATIONS OF THE FIRST DEGREE 79 

tramsposing and reducing 

30a; = 333 : 
Whence, by dividing both members of the equation by 30, 

x= 11.1. 
If we substitute this value of x, for x, in the given equation, 
it will verify it, that is, make the two members equal to eacll 
other. 

Find the value of x in each of the following 

EXAMPLES. 

1. 3a; — 2 + 24 = 31. Ans. x = S, 

2. a; + 18 = 3a; — 5. Ans, x = 11^. 

3. 6 •— 2ar -f 10 = 20 — 3af — 2. Ans. x =z 2 

4. a; + -— a; + — a; = 11. Ans. a; = 6. 

2 3 

1 o 

5. 2x a; + 1 = 5aj — 2. Ans. a; = — , 

..«.«« T , 6 — 3a 

6. 3aa; + — 3 z= bx — a, Ans. x = -. 

2 6a — 26 

. x — S , X ^^a;— 19 . ^„- 

1. = 20 . Ans. X = 23i. 

2 3 2 * 

^a; + 3,a; ^ x — 5 . „« 

8. — = 4 — . Ans. X = 3t%. 

2 3 4 ^^ 

ax — 6 a &a; 6a; — a . 36 

9. i = • : Ans. X = 



4 3 2 3 3a — 26 

, ^ 3aa; 26a; , ^ , cdf + 4cc? 

10. 4 =/. Ans. X = /, ^, . 

c d -^ Sad — 2bc 

^^ Sax^h 36 — c , , , 56 + 96 — 7c 

11. J- — = 4 — 6. Ans. X = — 

1 2 16a 

,^ a; a; — 2 , a; 13 

12. -— — — [- — = —-, ^dns, X = 10. 

5 3 2 3 



80 ELEMENTS OF ALGEBRA. [CHAP. IV 

.^ X XX X ahcdf 

13. -\ r- =/. Ans, X = — — :^— — . 

abed bed — acd + aod — aoc 

3ar — 5 4a; — 2 

14. X -^ 1 — — = a; -f 1. Ans. a; = 6. 

lo / 11 

15. y - ^ - ^- = ~ 12||. Ans. X = 14. 



4a; -2 3a; - 1 

16. 2a; — = — - — . Ans. x = 3. 

5 2 

irv o . ^^ — ^ . >« 3a + <? 

17, Sx H — =x -\- a. A71S. x = -. 

_ g^ + 3a3^, + 4a252 _ Q^p ^ 25* 
''^' "^ ~ 26 (2a2 + a6 - 6^) ' . 

Problems giving rise to Equations of the First Degree^ involv- 
ing hut cr^e Unknown Quantity. 

81* The solution of a problem, by means of algebra, consists 
of two distinct parts — 

1st. The statement of the problem ; and 

2d. The solution of the equation. 

We have already explained the methods of solving the equar 
tion ; and it only remains to point out the best manner of making 
the statement. 

The statement of a problem is the operation of expressing, 
algebraically, the relations between the known and unknown 
quantities which enter it. 

This part cannot, like the second, be subjected to any well- 
defined rule. Sometimes the enunciation of the problem furnishes 
the equation immediately ; and sometimes it is necessary to dis- 
cover, from the enunciation, new conditions from which an equa 
tion may be formed. 



CHAP. IV. EQUATIONS OF THE FIRST DEGREE. 81 

The conditions enunciated are called explicit conditions^ aii<3 
those which arc deduced from them, implicit conditions. 

In almost all cases, however, we are enabled to discover the 
equation bj applying the following 

RULE. 

Denote the unknown quantity by one of the final letters of tlte 
alphabet^ and then indicate^ by means of algebraic signs, the same 
operations on the known and unknown quantities, as would be 
necessary to verify the value of the unknown quantity, were such 
value known. 

PROBLEMS. ^ 

1. Find a number such, that the sum of one half, one third 
and one fourth of it, augmented by 45, shall be equal to 448. 

Let the required number be denoted by X, 



X 
X 



Then, one half of it will be denoted by - - - - 
one third of it by - - - - 

one fourth of it bv — • 

J 4 ' 

XXX 

and by the conditions, it- + -^H 1-45 = 448. 

/^ o 4 

Transposing - - ^ -f 4- + 4" = 448 - 45 = 403 : 

♦clearing of fractions, - - - - Qx -}- 4x -{- Sx = 4836 ; 

reducing, 13a; = 4836 ; 

iience, x = 372. 

Let \is see if this value will verify the equation. We have, 

372 , 372 .372 _ _ 

■^ + j-^- 4- -^ 4- 45 = 186 -f- 124 + 93 + 45 = 448. 



82 ELEMENTS OF ALGEBRA. [CHAP. IV. 

2. What number is that whose third part exceeds its fourtli 
by 16? 

Let the required number be denoted by x. 

Then, — x will denote the third part ; 

o 

and -j-x will denote the fourth parL 

B}' the conditions of the problem, 

_^__.,^ 16. 

Clearing effractions, - Ax — Sx = 192; 
reducing, .... - a; = 192. 

Verification. 
192 192 ,^ 

or, - - - 16 = 16. 

3. Out of a cask of wine which had leaked away a third part, 
21 gallons were afterward drawn, and the cask was then half 
full : how much did it hold 1 

Suppose the cask to have held x gallons. 

X 

Then, - - - - -^ will denote what leaked away; 

o 

X 

and .... __ -j- 21 will denote what leaked out and 

o 

•also what was drawn out. 

By the conditions of the problem. 

Clearing effractions, - 2x + 126 = Sx ; 

reducing — x = — 126 ; 

dividing by — 1 - • x = 126. 

Yerification. 
or. 63 = 63. 



CHAP. IV.l EQUATIONS OF THE FIRST DEGREE. 83 

4. A fish was caught whose tau weighed 9/&. ; his head weighed 
as much as his tail and half his body ; his body weighed as much 
as his head and tail together : what was the weight of the fish "i 

Let - - 2x denote the weight of the body ; 
then - - 9 \- X A\dll denote weight of the head; 
and since the body weighed as much as both head and tail, 
2x = 9-}- 9 + x 
or, - 2a; — a; =:= 18 ; whence, x = 18. 

Verification, ^ 

2 X 18 - 18 = 18 ; or, 18 - 18. 

Hence, the body weighed -. - - SQlbs ; 

the head weighed 27 lbs ; 

the tail weighed - 9lbs ; 

and the whole fish 12lbs. 

5. A person engaged a workman for 48 days. For each day 
that he labored he received 24 cents, and for each day that he 
was idle, he paid 12 cents for his board. At the end of the 48 
days the account was settled, when the laborer received 504 
cents. Required the number of working days^ and the number of 
days he was idle. 

If these two numbers were known, by multiplying them re. 
spectively by 24 and 12, then subtracting the last product from 
the first, the result would be 504. Let us indicate these 
operations by means of algebraic signs. 

Let - - x denote the number of working days ; 

then 48 — x will denote the number of idle days ; 

24 X a; = the amount earned, and 
12 (48 — x) = the amount paid for his board. 
Then, from the conditions, 

24:x — 12 (48 — x) = 504 
or, 24.x - 57Q -^ 12x = 504, 

Reducing . 36a; = 504 -f 576 = 1080 

whence, a; = 30 the working days, 

and, 48 — 30 = 18 the idle days. 



84 ELEMENTS OF ALGEBRA. [CHAP. IV. 

Verification. 

Thirty days' labor,- at 24 cents a day 
imounts to - - ^ 30 X 24 = 720 cts ; 

and 18 days' board, at 12 cents a day, 
amounts to - - - - 18 X 12 = 21G cts ; 

and the amount received, is their diiFerence, 504 cts. 

The preceding is but a particular case of a general problem 
which may be enunciated as follows. 

•A. person engaged a workman for n days. For each day 
that he labored, he was to receive a cents, and for each day 
that he was idle, he was to pay h cents for his board. At 
the end of the time agreed upon, he received c cents. Re- 
quired the number of working days, and the number of idle 
days. 

Let - ~ X denote the number of working days ; then, 
n — X will denote the number of idle days ; 

ax will denote the number of cents he received; and 
b {n — x) will denote the number he paid out. 
From the conditions of the problem, 
ax — b {71 — x) z=z c. 
Performing the indicated operations, transposing and factoring, 
we find, 

[a -\- b)x z= c ~\- bn, 

whence, x= -^j-) the number of working days ; and 

n — X ~ ;— -. the number of idle days. 

a+ b' '' 

If we make ?i = 48, a = 24, 5 = 12 and c — 504, we obtain, 

504 + 576 

X ^ .— — 30 ; and 48 — .r = 18 ; as before found. 

C. A fox, pursued by a greyhound, has a start of 60 leaps. 
He makes 9 leaps while the greyhound makes but 6; but 3 
leaps of the greyhound are equivalent to 7 of the fox. How 
many leaps must the greyhound make to overtake the fox? 



CHAP. IV.J EQUATIONS OF THE FIRST DEGREE. 85 

Let us take one of the fox leaps as the unit of distance-, 

then, 3 leaps of the greyhound being equal to 7 leaps of the 

7 
fox, one of the greyhound leaps will be equal to — . 

o 

Let X denote the number of leaps the greyhound must make 
before overtaking the fox. 

Then, since the fox makes 9 leaps while the hound makes (J, 
9 3 

r ^^ ¥" 

will denote the number of leaps the fox makes in the same time. 

7 

— X will denote the whole distance passed over by the hound ; 

— X will denote the whole distance passed over by the fox. 
Then, from the conditions of the problem, 

Clearing of fractions, 14a; = 360 + 9a;, 

transposing and reducing, 5a: = 360, 
whence, ic = 72 ; 

3 3 

and — rr = — - X 72 = 108, the nu>txber of fox le^ps, 

lit /i 

Verification. 

^ X 72 ^^ , 3 X 72 
—^ - 60 + -^- ; 

or, ... - 168 = 168. 

7. A can do a piece of work alone in 10 days, and B in 13 

days : in what time can they do it if they work tog^tht^c ? 

Denote the number of days by x^ and the work to ^' ^^>ne 

ly 1. Then, in 

• 1 

1 day A can do — of the work ; and in 

1 day B can do — of the work ; hew^^e, h 

lo 

X 

X days A can do — of the *7ork ; and \\> 

X 

X days B can do — of the work : 

lo 



86 ELEMENTS OF ALGEBRA. ICHAP. IV. 

Hence, hj the conditions of the question, 

clearing of fractions, 13a; + 10a; == 130 : 

hence, a; = 5|f , the number of days. 

8. Divide $1000 between A, B and C, so that A shall have 
$T2 more than B, and C $100 more than A. 

A71S, A's share = $324, B's =r $252, C's = $424. 



9. A and B play together at cards. A sits down with 
and B with $48. Each loses and wins in turn, when it ap- 
pears that A has five times as much as B. How much did A 
win? Ans. $26. 

10. A person dying, leaves half of his property to his wife, 
one sixth to each of two daughters, one twelfth to a servant, 
and the remaining $600 to the poor : what was the amount 
of his property? Ans. $7200. 

11. A father leaves his property, amounting to $2520, to four 
sons. A, B, C and D. C is to have $360, B as much as C 
and D together, and A twice as much as B less $1000: how 
much do A, B and D receive? 

Ans. A $760, B $880, D $520. 

12. An estate of $7500 is to be divided between a widow, two 
sons, and three daughters, so that each son shall receive twice "as 
much as' each daughter, and the widow herself $500 more than 
all the children: what was her share, and what the share of 
each child ? r Widow's share, $4000. 

Ans. } Each son, $1000. 

( riach daughter, $500. 

13. A company of 180 persons consists of men, women and 
children. The men are 8 more in number than the women, and 
■Jxe children 20 more than the men and women together : how 
many of each sort in the company ? 

Ans. 4A men, 36 women, 100 children. 



CHAP. IV.J EQUATIONS OF THE FIRST DEGREE. 87 

14. A father divides $2000 among five sons, so that each elder 
should receive $40 more than his next younger brother : what is 
the share of the youngest? Ans, $320. 

15. A purse of $2850 is to be divided among three persons, 
A, B and C; A's share is to be ^j of B's share, and C is to 
have $300 more than A and B together : what is each one's 
share? Ans. A's $450, B's $825, C's $1575. 

16. Two pedestrians start from the same point ; the first steps 
twice as far as the second, but the second makes 5 steps while 
the first makes but one. At the end of a certain time they are 
'iOO feet apart. Now, allowing each of the longer paces to* be 8 
fvet, how far will each have traveled ? 

Ans. 1st, 200 feet; 2d, 500. 

17. Two carpenters, 24 journeymen, and 8 apprentices, re- 
ceived at the end of a certain time $144. The carpenters 
received $1 per day, each journeyman half a dollar, and each 
apprentice 25 cents: how many days were they employed?' 

' Ans. 9 days. 

18. A capitalist receives a yearly income of $2940 ; four fifths 
of his money bears an interest of 4 per cent., and the remainder 
of five per cent. : how nauch has he at interest ? 

Ans. $70000. 

19. A cistern containing 60 gallons of water has three unequal 
cocks for discharging it ; the largest will empty it in one hour, 
the second in two hours, and the third in three : in what time 
will the cistern be emptied if they all run together ? 

Ans. 32jY mm. 

20. In a certain orchard | are apple-trees, J peach-trees, 
1- plum-trees, 120 cherry-trees, and 80 pear-trees : how many 
trees in the orchard ? Ans. 2400. 

21. A farmer being asked how many sheep he had, answered 
that he had them in five fields in the 1st he had i, in the 
2d -^, in the 3d -J-, in the 4th ^^, and in the 5th 450 : how 
many had he? Ans. 1200. 



88 ELEMENTS OF ALGEBRA. LCHAP. IV. 

22. My horse and saddle together are worth $132, and the 
horse is worth ten times as much as the saddle : what is the 
value of the horse? A?is. $120. 

23. The rent of an estate is this year 8 per cent, greater thau 
it was last. Tliis year it is $1890 : what was it last year ] 

Ans.-U^bO, 

24. What number is that from which, if 5 be subtracted, |- of 
the remainder will be 40 1 Ans. 65. 

25. A post is i in the mud, -i- in the water, and ten feet above 
the water : what is the whole length of the post 1 

Ans. 24 feet. 

26. After paying -J and J of my money, I had Q6 guineas le^ 
in my purse : how many guineas were in it at first 1 

Arts. 120. 

27 A person was desirous of gi^'ing 3 pence apiece to some 
beggars, but found he had not money enough in his pocket by 8 
pence ; he therefore gave them each two pence and had 3 pence 
remaining: required the number of beggars. Ans. 11. 

28. A person in play lost J of his money, and then won 3 
sliillings ; after which he lost J of what he then had ; and this 
done, found that he had but 12 shillings remaining : what had 
he at first? Ans. 20s. 

29. Two persons, A and B, lay out equal sums of money in 
trade ; A gains $126, and B loses $87, and A's money is new 
double B's : what did each lay out 1 A^is. $300. 

30. A person goes to a tavern with a certain sum of money 
m his p( >cket, where he spends 2 shillings ; he then borrows 
as much money as he had left, and going to another taverD, 
lie there spends 2 shillings also ; then borrowing again as 
nmch money as was left, he went to a third tavern, where, 
likewise, he spent 2 shillings and borrowed as much as he 
liad left ; and agam spending 2 shillings at a fourth taveru, 
he then had nothmg remainincr. What had he at first? 

Ans. Ss. 9d. 



CHAP. III.] EQUATIONS OF THE FIRST DEGREE. 89 

31. A farmer bought a basket of eggs, and offered them at 7 
cents a dozen. But before he sold any, 5 dozen were broken 
by a careless boy, for which he was paid. He then sold the re- 
mainder at 8 cents a dozen, and received as much as he would 
have got for the whole at the first price. How many eggs had 
he in his basket *? Ans. 40 dozen. 

Equatio7is of the First Degree involving more than one 
Unhnown Quantity. 

82» If we have an equation between two unknown quantities, 

we may find an expression for one of them in terms of the 

other and known quantities ; but the value of this unknown 

quantity could only be determined by assuming a value for 

the second. Thus, from the equation, 

^ + 2y = 4, 
we may deduce 

x = 4. -2y, 

but cannot find a value for x without assuming one for y. 

If, however, we have another equation between the two un 
known quantities, the values of these quantities being the same 
in both, we may find, as before, an expression for x in terms 
of ?/, and this expression placed equal to the one already 
found, will give an equation containing but one unknown quan- 
tity. Let us take 

a; + 3y = 5, 
from which we find 

x z=z 5 — oy. 

If we place this expression equal to that before found, we 

deduce the equation 

4 - 2y = 5 - 3y, 
fi'om the solution of which we find, ?/ = 1. 

This value of y, substituted in either of the given equatious, 
gives X — 2: hence, 

X =z2 and y = 1 satisfy both equations. 

We see that in order to find determinate values for two 
unknown quantities, we must have two independent equations. 
Simultaneous equations are those in which the values of the 
unknown quantities are the satue in them all at the same time 



90 ELEMENTS OF ALGEBRA, [CHAP. IV. 

In the same manner it may be shown that tc determine the 
values of three unknown quantities, we must have three equa- 
tions ; and generally, to determine the values of n unknown 
quantities we must have n equations. 

Elimination. 

83* Elimination is the operation of combining several equation» • 
involving several unknown quantities^ and deducing therefrom a less 
number of equations involving a less number of unkiiown quantities. 

There are three principal methods of elimination : 

1st. By addition or subtraction. 

2d. By substitution. 

3d. By comparison. 

We shall explain these methods separately. 

Elimination hy Addition or Subtraction. 

84. Let us take the two equations 
4x — by = 5, 
nx-{-2gz= 21. 
If we multiply both members of the first equation by 2, 
the co-efficient of y in the second, and both members of the 
second equation by 5, the co-efficient of y in the first, we obtain, 
Sx - lOy = 10, 
15x + lOy = 105 ; 
in which the co-efficients of y are numerically the same in both. 
If, now, we add these equations member to member, we find 

2Sx = 115. 
In this case y has been eliminated by addition. 
Again, let us take the equations 

2x-^Sy= 12, 
Sx + 4.y= 17. 
If we multiply both members of the first equation by 8, 
the co-efficient of a; in the second, and multiply both mem- 
bers of the second equation by 2, the co-efficient of x in the 

first, we shall have, 

6x ^9y = 36, 
• 6x-\-8y = S4; 



CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 91 

in which the co-efficients of x are the same in both. If, now, 

we subtract the second equation from the first, member from 

member, we find, 

y = 2. 

Here, x has been eliminated hy subtraction. 

In a similar naanner we may eliminate one unknown quantit}' 
between any two equations of the first degree containing any 
number .of unknown quantities. The rule for elimination by 
addition and subtraction may be simplified by using the least 
common multiple. Hence, for elimination by addition or sub- 
traction, we have the following 

RULE. 

Prepare the two equations in such a manner that the co-efficients 
of the quantity we wish to eliminate shall he numerically equal 
in both : then, if the two co-efficients have contrary signs, add the 
equations^ member to member ; if they have the same sign, sub- 
tract them member from member, and the resulting equation will 
be independent of that quantity. 

Elimination hy Suhstitution. 

85. Let us take the equations, 

6x-i-7y = 4S, and Ux + 9y — 69. 
Find, from the first equation, the value of x in terms of y, 
which is, 

43 - 7y 
^ = —5-- 
Substitute this value for x in the second equation, and wa 
shall have, 

ll^(^^2^ + 9,.e9; or, 

reducing, - - . 473 - 77?/ + 45y == 345. 

In a similar manner Ave may eliminate one unknown quantity 
between two equations of the first , degree containing any number 
of unknown quantities. 

Hence, for eliminating by substitution, we have the following 



92 ELEMENTS OF ALGEBRA. [CHAP. IV. 

RULE. 

Find from one equation the value of the unJcnown quantity to 
he eliminated in terms of the others : substitute this value in the 
other equation for the unknown quantity to he eliminated^ and the 
resulting equation will he independent of that quantity. 

Elimination hy Comparison. 



86. Let us take 


the 


equations, 
5;r 4- 7y rz 43, 
11:?; + 9y = 69. 




Finding the value of 


X in terms of y, 


from both equations 


we have. 




69 -9y 





11 

If, now, we place these values equal to each other, we shall have, 
43 - 7y _ 69 - 9y 

5 ~ ir~' 

reducing, - - - 473 — 77y = 345 — 45y. 

Here, x has been eliminated. Generally, if we have two 
equations of the first degree containing any number of unknown 
quantities, any one of them may be eliminated by the following 

RULE. 

Find the value of the quantity we wish to eliminate, in terms 
of the others, from each equation, and then place these values 
equal to each other : the resulting equation will he independent 
of the quantity whose values were found. 

The new equations which arise, fr-om the two last methods 
of elimination, contain fractional terms. This inconvenience is 
avoided in the first method. Tlie method hy substitution is, 
however, advantageously employed whenever the co-efiicient of 
either of the unkno^ni quantities in one of the equatioQs is equal 
to ], because then the inconvenience of which we have just 



CHAP. IV.J EQUATI0:5rS OF THE FIKST DEGREE. 93 

spoken doe. not c^cur. We shall sometimes have occasion to 
employ this method, but generally the method by addition and 
subtraction is preferable. When the co-efficients are not too 
great, the addition or subtraction may be performed at the 
same time with the multiplication that is made to render the 
co-efficients of the same unknown quantity equal to each other. 

Thore is also a method of elimination by means of thts 
greatest common divisor, which will be explained in its appro- 
priate place. 

87* Let us now consider the case of three equations involving 
three unknown quantities. 

^ 5a; — 6y -}- 4^ = 15, 

Take the equations, •< 7a; + 4y — 00 = 19, 
(2x-{- y + 6;2=:46. 

To eliminate z from the first two equations, multiply the first 
equation by 3 and the second by 4 ; and since the co-efficients 
of z have contrary signs, add the two results together : this gives 
a new equation, - - - - 43a; — 2?/ == 121 

Multiplying both miembers of the second 
equation by 2, a factor of the co-efficient of 
z in the third equation, and adding them, 
member to member, we have - - - 16a; -j- 9y = 84.^ 

The question is then reduced to finding the values of x and y, 
which will satisfy these new equations. 

Now, if the first be multiplied by 9, the second by 2, and 
the results be added together, we find 

419a; = 1257, whence a; = 3. 

By means of the two equations involving x and y, we may 
determine y as we have determined x ; but the value of y may 
De determined more simply, since by substituting for x its 
value found above, the last of the two equations becomes, 
48 + 9y = 84, whence y = 4. 

In the same manner, by substituting the values of x and y, 
fclie first of the three proposed equations becomes, 
15 — 24 -f 4;? = 15, whence z = Q, 



94 ELEMENTS OF AUtiLBUX. [CHAP. IV, 

If we have a group of m simultaneous equations containing m 
unknown quantities, it is evident, from principles already ex- 
plained, that the values of these unknown qaantities may be 
found by the following 

RULE. 

I. Combine one of the m equations with each of the m — \ others, 
separately, eliminating the same unknown quantity ; there will result 
m — 1 equations containing m — 1 unknown quantities, 

n. Combine one of these with each of the m — 2 others^ s^a- 
rately, eliminating a second unknown quantity ; there will result 
m — 2 equations containing m — 2 unknown quantities. 

Til. Continue this operation of combination and elimination, till 
we obtain, finally, one equation containing one unknown quantity. 

IV. Find the value of this unknown quantity by the rule for 
solving equations of the first degree containing one unknown quan- ^ 
tity : substitute this value in either of the two preceding equations 
containing two unknown quantities, and determine the value ' of a 
second unknown quantity : substitute these two values in either of 
the three equations involving three unknown quantities, and so on 
till we find the values of them all. y 

It often happens that some of the proposed equations do not***^ 
contain all the unkno^vn quantities. In this case, with a little 
address, the elimination is very quickly performed. 

Take the four equations involving four unknown quantities, 

2x-^y-\-2z = l^ . (1) 4y + 20 = 14 - (3). 

4u-2x = S0 - (2) 5y-{-Su=S2 - (4). 

By examining these equations, we see that the elimination of 
z in equations (1) and (3), will give an equation involving x 
and y ; and if we eliminate u in the equations (2) and (4), we 
shall obtain a second equation, involving x and y. In the first 
place, the elimination of z, in (1) and (3) gives 7y — 2x = 1 - (5), 
that of u, in (2) and (4), gives - - 20y-\-6x = SS- (6). 

From (5) and (6) we readily deduce the values of y = 1 and 
a; = 3 ; and by substitution in (2) and (3), we also find m = 9 
and z = 6. 



1 



CHAP. IV.J EQUATIONS OF THE FIRST DEGREE. 

EXAMPLES. 



95 



1. Given 2^5 + Sy = 16, and 3a; — 2y = 11 to find the values 
of X and y. , Ans, a; = 5, y = 2. 

2. Given y + f = ^0' '^^ T + 1^ = 120 '" ^^^ *^" 



values of x and y. 



^--:=-i, . = |. 



S. Given — + Ty = 99, and — + 7a; = 51 to find the values 



pf X an-d y. 



Ans. X = 7, y = 14. 



4. Given f- 12 = f + 8, and ^^ + f- 8 =^-% 27 
to find the values of x and y. ^^5. a; = 60, y = 40. 

a;+ y+ ' = 29 
x+ 2y + 30 = 62 



6. Given 



^. i''+h+i' = '^^ 



to find a;, y, and gr. 



6. Given 



Ans. a; = 8, y = 9, = 12. 
2a; + 4y — 80 = 22 j 

4a; — 2y + 50 = 18 > to iuid x, y, and «. 
6a; + 7y - = 63 ) 

Ans. X = S, y = 7, = 4. 



7 Given 



^+yy + y^ = 32 

1,1 1 

y^+X2^ + y. = 15 



to find a;, y, and 2. 



8. Given 



^725. a; = 12, y = 20, = 30. 

r 7a; - 20 + 3?* = 17 

4y— 20+ ^=11 

5y— 3a;— 2u = 8 

4y — Su-^ 2t = '9 

30 + 82^ = 33 J 
Ans. x=z2, y = 4, = 3, w = 3, ^ = 1 



>■ to find a;, y, e, w, 
and ^. 



96 ELEMENTS OF ALGEBRA. [CHAP. IV. 

PROULEMS GIVING RISE TO SIMULTANEOUS EQUATIONS OF THE I=IRST 

DEGREE. 

1. What fraction -is that, to the numerator of which, if 1 be 
added, its value will be one third, but if 1 be added to its 
ienominator, its value will be one fourth? 
Let X denote the numerator, and 

y the denominator. 
From the conditions of the problem, 
x + \ \ 





y 


-y 




X 


1 




y + 1" 


~ 4' 


Clearing of fractions, 


the first 


equation gives, 




3^ + 3- 


= y, 


and the 2d, 


4^ 


= 2/+l. 


Whence, by eliminating y, 






x-'6^ 


= 1, 


and 


X ~ 


r4. 


Substituting, we find, 






■\ .1 '-an 


y = 


.15; 
4 



and the required fraction is — . 

io 

2. To find two numbers such that their sum shall "He equal 
to a and their difference equal to 6. 
Let x denote the greater number, and 

y the lesser number. 
From the conditions of the problem, 
x + y =a, 
X — y ^^b. 
Eliminating y by addition, 

2x =z a -\- b, 
a . b 



By substitution, 



^~ 2 2 



CHAr. IV.J EQUATIONS OF THE FIRST DEGREE 97 

3. A person possessed a capital of 30000 dollars, for which 
he drew a certain interest per annum ; but he owed the sum 
of 20000 dollars, for which he paid a certain interest. The 
interest that he received exceeded that which he paid by 800 
dollars. Another person possessed 35000 dollars, for whicli 
he received interest at the second of the above rates ; but he 
owed 24000 dollars, for which he paid interest at the first 
of the above rates. The interest that he received exceeded 
that which he paid by 310 dollars. Required the two rates 
of interest. 

Let X denote the first rate, and 

y the second rate. 

Then, the interest on $30000 at x per cent, for one year will be 

$30000^; ^„^^ 

-j^^ or $300^. 

The interest on $20000 at y per cent, for one year will be 

$20000y ^^^^ 

-^ or $200y. 

Hence, Irom the first condition of the problem, 
300a; — fiOOy = 800 ; 
or, - - - - Sx— 2y r=r 8 - - - (1). 
In like manner fi'om the second condition of the problem we find 

353/- 24^= 31 - - - (2). 
Combining equations (1) and (2) we find, 

y = 5 and x = 6. 
Hence, the first rate is per cent, and the second rate 5 
per cent. 

Verification. 

$30000, placed at 6 per cent , gives $300 x = $1800. 
$20000 do 5 do $200 x 5 = $1000. 

. And we have 1800 — 1000 = 800. 

The second condition can be verified in the same manner. 

4. Tiiere are three ingots formed by mixing together three 
metals in different proportions. 

7 



98 ELEMENTS OF ALGEBRA. [CHAP. IV 

One pound of the first contains 1 ounces of silver, 3 ounces 
of copper, and 6 ounces of pewter. 

One pound of the second contains 12 ounces of iilver, 3 ounces 
of copper, and 1 ounce of pewter. 

One pound of the third contains 4 ounces of silver, 7 ouncer 
of copper, and 5 ounces of pevvter. 

It is reqiired to form from these three, 1 pound of a fourth 
ingot which shall contain 8 ounces of silver, 3} ounces of cop- 
per, and 4i ounces of pewter. 

]jet X denote the number of ounces taken from the first. 

y denote the number of ounces taken from the second. 
z denote the number of ounces taken from the third. 

Now, since 1 pound or 16 ounces of the first ingot contains 7 

ounces of silver, one ounce will contain -— of 7 ounces : that 

lo 

is, — ounces ; and 

7x 
X ounces will contain — - ounces of silver. 
Id 

. 12y ^ .-, 

y ounces will contain --— ounces oi silver, 
^ 16 

4^ 
z ounces ^vill contain — - ounces of silver, 
lb 

But since 1 pound of the new ingot is to contain 8 ounces of 

silver, Ave have 

Ix 12y 4^ _g. 
16 ^ 16 "^ 16 
4)r, clearing of fi-actions, we have, 
for the silver, 7^ + 12?/ + 4^ = 128 ; 

for the copper, S.r + 3?/ + 7^ = 60 ; 

und for the pewter, ^x -\- y -\- 5z z=z 68. 

Whence, finding the values of x, y and z, we have 
a: =: 8, the number of ounces taken from the first. 
2/ = 5 " " 

z = S 'i 

5. What two numbers are they, whose sum is 33 and whose 
difference is 71 Ans. 20 and IS. 



(( 


u 


u 


" second. 


« 


11 


u 


" third. 



CHAP. IV. J EQUATIONS OF THE B^IRST DEGKEE. 99 

6. Divide the number 75 into two such parts, that three times 
the greater may exceed seven times the less by 15. 

Ans. 54 and 21. 

7. In a mixture of wine and cider, J of the whole plus 25 
gallons was wine, and ^ part minus 5 gallons, was cider ; how 
many gallons were there of each 1 

A71S. 85 of wine, and 35 of cider. 

8. A bill of £120 was paid in guineas and moidores, and the 
number of pieces of both sorts that were used was just 100 ; if 
the guinea were estimated at 21s., and the moldore at 27s., how 
many were there of each? A71S, 50. 

9. Two travelers set out at the same time from London and 
York, whose distance apart is 150 miles ; they travel toward 
each other; one of them goes 8 miles a day, and the other 
7; in what time will they meet? Ans. In 10 days. 

10. At a certain election, 375 persons voted for two candi 
dates, and the candidate chosen had a majority of 91 ; how 
many voted for each 1 

Ans. 233 for one, and 142 for the other. 

11. A's age is double B's, and B's is triple C's, and the sum 
of all their ages is 140 ; what is the age of each '? 

Ans. A's = 84, B's = 42, and C's = 14. 

12. A person bought a chaise, horse, and harness, for £60 ; 
the horse came to twice the price of the harness, and the chaise 
to twice the price of the horse and harness ; what did he give 
for each 1 ^ £13 6s. 8c?. for the horse. 

Ans. <£ 6 13s. M. for the harness. 
' £40 for the chaise. 

13. A person has two horses, and a saddle worth £50 ; now, 
if the saddle be put on the back of the first horse, it will make 
his value double that of the second ; but if it be put on the back 
of the second, it will make his value triple that of the first 
what is the value of each horse ? 

Ans. One £30, and the other £40. 



100 ELEMENTS OF ALGEBRA. [CHAP. [V 

14. Two persons, A and B, have each the same income. A 
isaves I of his yearly; but B, by spending £50 per annum mere 
t^an A, at the end of 4 years finds himself £100 in debt; what 
is the income of each 1 Ans. £125. 

15. To divide the number 36 into three such parts, that ^ of 
the first, -J of the second, and ^ of the third, may be all equaJ 
to each other. • Ans. 8, 12, and 16. 

16 A footman agreed to serve his master for £8 a year and 
A livery, but was turned away at the end of 7 months, and re 
reived only £2 135. 4d. and his livery ; what was its value 1 

Ans. £4 165. 

17. To divide the number 90 into four such parts, that if the 
first be increased by 2, the second diminished by 2, the third 
multiplied by 2, and the fourth divided by 2, the sum, difference, 
product, and quotient, so obtamed, will be all equal to each other. 

Ans. The parts are 18, 22, 10, and 40. 

18. The hour and minute hands of a clock are exactly together 
at 12 o'clock ; when are they next together ? 

Ans. 1 h. 5^j min. 

19. A man and his wife usually drank out a cask of beer in 
12 days ; but when the man was from home, it lasted the woman 
SO days; how many days would the man be in drmking it 
alone? Ans. 20 days. 

20. If A and B together can perform a piece of work in 8 
days, A and C together in 9 days, and B and C in 10 days; 
how many days would it take each person to perform the same 
work alone 1 Ans. A 14f| days, B 17f f , and C 233^. 

21. A laborer can do a certain work expressed by a, in a time 
expressed by 5; a second laborer, the work c in a time d; a 
third, the work e in a time /. Required the time it would ta>ke 
the three laborers, workir-g together, to perform the work g, 

hdfg 



Ans. 



'adf -^ bcf -\- bde 



CHAP. IV.J EQUATIONS OF THE FIRST EEGREE. 10) 

22. If 32 pounds of sea water contait 1 pound of salt, how 
much fresh water must be added to these 32 pounds, m order 
that the quantity of salt contained in 32 pounds of the new mix- 
ture shall be reduced to 2 ounces, or -i of a pound'? 

Ans. 224 lbs. 

23. A number is expressed by three figures ; the sum of these 
figures is 11 ; the figure in the place of units is double that in 
the place of hundreds; and when 297 is added to this number, 
the sum obtained is expressed by the figures of this number re- 
versed. What is the number ? Ans. 326. 

24. A person who possessed 100000 dollars, placed the- greater 
part of it out at 5 per cent, interest, and the other part at 4 per 
cent. The interest which he received for the whole amounted 
to 4^40 dollars. Kequired the two parts. 

„ Ans. 164000 and- $36000. 

25. A person possessed a certain capital, which he placed out 
at a certain interest. Another person possessed 10000 dollars 
more than the first, and putting out his capital 1 per cent, more 
advantageously, had an income greater by 800 dollars. A third, 
possessed 15000 dollars more than the first, and putting out his 
capital 2 per cent, more advantageously, had an income greater 
by 1500 dollars. Required the capitals, and the three rates of 
interest. 

Sums at interest, $30000; $40000, $45000. 

Rates of interest, 4 5 6 per cent. 

26. A cistern may be filled by three pipes, A, B, C. By 
the two first it can be filled in 70 minutes ; by the first and 
third it can be filled in 84 minutes ; and by the second and 
third in 140 minutes. What time will each pipe take to do 
it in ? What time will be required, if the three pipes run 
together 1 

r A in 105 minutes. 

Ans. -J B in 210 minutes. 

' C in 420 minutes. 
All will fill it m one hour. 



102 ELEMENTS OF ALGEBRA. [CHAP. IV 

27. A, has 3 purses, each contaimng a certain sum of money. 
If $20 be taken out of the first and put into the second, it 
will contain four times as much as remains in the first. If $60 
be taken from the second and put into the third, then this will 
contain IJ times as much as there remains in the second. Again, 
if $40 be taken from the third and put into the first, then 
the third will contain 2| times as much as the first. What 
were the contents of each purse 1 r 1st. $120. 

Ans. yzd. $380. 
(3d. $500. 

28. A banker has two kinds of money; it takes a pieces of 

the first to make a crown, and b of the second to make the 

same sum. Some one offers him a crown for e pieces. How 

many of each kind must the banker give him ? 

1 • -. ci(c — b) ^ -, 1 . -, b (a — c) 

Ans. 1st kind, -^^ -^ : 2d land, -^^ -^. 

a — b a — b 

29. Find what each of three persons. A, B, C, is worth, 
knowing, 1st, that what A is worth added to I times what B 
and C are worth, is equal to ^ ; 2d, that what B is worth 
added to m times what A and C are worth, is equal to q ; 
3d, that what C is worth added to 71 times what A and B are 
worth, is equal to r. 

If we denote .by s what A, B, and C, are worth, we intro- 
duce an auxiliary quantity, and resolve the question in a very 
simple manner. • '; - > 

30. Find the values of the estates of six persons, A, B, C, D, 
E, F, from the following conditions : 1st. The sum of the estates 
of A and B is equal to a : that of C and D is equal to b ; and 
that of E and F is equal to c. 2d. The estate of A is worth m 
times that of C ; the estate of T> is worth 71 times that of E, and 
the estate of F is worth p times that of B. 

This problem may be solved by means of a single equation, 
involving but one unkno^vn quantity. 



CHAP. IV.'l EQUATIONS CF THE FIRST DEGREE. 108 

Of Indeterminate Equations and Indeterminate Problems. 

88. x\n equation is said to be indeterminate ^^heii it may be 
satisfied for an infinite number of sets of values of the unknown 
quantities which enter it. 

Every single equation containing two unknown quantities is iiide- 
terminate. 

YoT example, let us take the equation 
6x — Sy = 12, 



12 + 3y 

whence, - - x = . 








Now, by niaking successively. 








y=h 2, 3, 4, 


5, . 


6, 


&c^ 


18 21 24 

^ = ^' ¥' y y 


27 
5' 


6, 


&c.. 



and any two corresponding values of x, y, being substituted m 
the given equation, 

5x — Sy = 12, 
will satisfy it : hence, there are an infinite number of values fur 
X and y w^hich will satisfy the equation, and consequently it is 
indeterminate; that is, it admits of an infinite number of solutions. 
If an equation contains more than two unknown quantities, we 
may find an expression for one of them in terms of the others. 
If, then, we assume values at pleasure for these others, we 
can find from this equation the corresponding values of the first ; 
and the assumed and deduced values, taken together, will satisfy 
the given equation. Hence, 

Every equation involving more than one unknown quantity is 
indeterminate. 

In general, if we have n equations involving more than n 
unknown quantities, these equations are indeterminate ; for we 
may, by combination and elimina';ion, reduce them to a single 
equation ontaining more than one unknown quantity, which we 
have already seen is indeterminate. 

If, on the contrary, we have a greater number of equatioDS 
than we have unknown quantities, they cannot all be satisfied 



104 ELEMENTS OF A.LGEBEA. [CH4P. IV 

unless some of them are dependent upoii the others. If we 
combine them, we may eliminate all the unknown quantities, and 
the resulting equations, which will then contain only known 
(quantities, will be so many equations of condition^ which must be 
satisfied in order that the given equations may admit of solution. 
For example, if we have 

X -{- y =za, 

^ — y = c, 

xy —d\ 

we may combine the fii'st two, and find, 

a , c ^ a c 

" = 3 + 2 ^"^ 2/=3-3; 

and by substituting these in the third, we shall find 

4-4=^' 
which expresses the relation between a, c and cZ, that must exist, 
in order that the three equations may be simultaneous. 

88*. A Prohlem is indeterminate when it admits of an infinite 
number of solutions. This will always be the case wh^n its 
enunciation involves more unknown quantities than the.-e are 
given conditions ; since, in that case, the statement of the p:*obleni 
will give rise to a less number of equations than there are 
unknown quantities. 

1st. Let it be required to find two numbers such jhat 5 
times the first diminished by 3 times the second s*7all be 
equal to 12. 

If we denote the numbers by x and y, the condition j of thft 
problem will give the equation 

5^ - 3y = 12, 
which we have seen is indeterminate: — Hence, the ^ .otlern 
admits of an infinite number of solutions, or is indete'r- 'mxte. 

2. Find a quantity such that if it be multiplied by a and 
the product increased by 6, the result will be equal to c time* 
the quantity increased by d. 



CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 105 

Let X denote the required quantity. Then froin the condition, 

ax ,-}- 6 z= ex + c?, 

d-h 

whence, - . . a; = . 

a — c 

If now we make the suppositions that d =zh and a nr- c, the 

value of X becomes -, wliich is a symbol of indetermination. 

If we make these substitutions in the first equatio*^, it be 
comes 

ax -^ b zzz ax + &, 
an identical equation [Ai't. 75), which must be satisfied for all 
values of x. ' These suppositions also render the conditions of 
the problem so dependent upon each other, that any quantity 
whatever will fulfil them all. 

Hence, the result - indicates that the problem admits of an 

infinite number of solutions. 

3. Find two quantities such that a times the first increased 
by b times the ' second shall be equal to c, and that d times 
the first increased by / times the second shall be equal to g. 

If we denote the quantities by x and y, we shall have from 
the conditions of the problem, 





ax -\- by - 


:zC, - . 


. . „ 


(1) 










dx+fy^ 


--9-> - ■ 


-■ . - 


(2) 








whence - 


cd — ag 
y -bd- af 


and r 


''-bd 


-<-/ 
-«/• 








If. now we 


make 
















cd = a.g, (3) 


and 


af = 


bd, (4) 


1 




, 


we shall find 


by multiplying these 


equations tog 


;ethep, 


membei 


by member, 




c/= hg. 












These suppositions, reduce the values of both x 


and y 


to 



0' 


-Fiom (3) ^ 


vve find, 














d 


= — , and from (4) 
c 


a. 


Xd = 









108 ELEMENTS OF ALGEBEA. LCHAP. IV. 

which 'v,v Dstit ited in equation (2), reduce it to 

ax -X- by z=z c, 
an eqv.atior which is the same as the first. 

Under this supposition, we have in reality but one equation 
between two unknown quantities, both of which ought to be inde- 
terminate. This supposition also renders the conditions of the 
problem so dependent upon each other, as to produce a less 
number of independent equations than there are unknown quan- 
tities. 

* . 

Generally, the result — , with the exception of the case men- 
tioned in Art. 71, arises from some supposition made upon the 
quantities entering a problem, which makes one or more condi- 
tions so dependent upon the others as to give rise to one or 

more indeterminate equations. In these cases the result — is 

a true answer to the probism, and is to be interpreted as 
indicating that the problem admits of an infinite number of 
solutions. 

Interpretation of Negative Results. 

89. From the nature of the signs -f- and — , it is clear that 
the operations which they indicat.c are diametrically opposite to 
each other, and it is reasonable to infer that if a positive re- 
sult, that is, one affected by the sign +, is to be interpreted 
m a certain sense, that a negative result, or one affected by 
./he sign — , should be interpreted in exactly the contrary 
sense. 

To show" that this inference is correct, wc shall discuss one 
or two problems giving rise to both positive and negative 
results. 

1. To find a number, which added to the m;mber 5, will 
^ive a sum equal to the number a. 

Let X denote the required number. Then from iho ctfuditions 
X -\- h z=z a^ V hence, x -^^ a — h. 



CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 107 

This formula will give the algebraic value of x in all the 
particular cases of the problem. 

For example, let a = 47 and 6 =: 29 ; 
then, re = 47 - 29 = 18. 

Again, let a = 24 and Z> = 31 ; 

then, X ^ 24 - 31 zzr - 7. 

This last value of a;, is called a negative solution. How is it 
to be interpreted? 

If we consider it as a purely arithmetical result, that is, as 
arising from a series of operations in which all the quantities 
are regarded as positive, and in which the terms add and sub- 
tract imply, respectively, augmentation and diminution, the prob- 
lem Avill obviously be impossible for the last values attributed 
to a and h ; for, the number b is already greater than 24. 

Ck)nsidered, however, algebraically, it is not so ; for we have 
found the value of a; to be — 7, and this number added, in the 
algebraic sense, to 3 J, gives 24 for the algebraic sum, and there- 
fore satisfies both the equation and enunciation. 

2. A father has lived a number of years expressed by a ; his 
son a number of years expressed by b. Find in how many years 
the age of the son will be one fourth the age of the father. 

Let X denote the required number of years. 

Then, a ~\- x will denote the age of the father j at the end of the 

and b -\- X will denote the age of the son ) required time. 

Hence, from the conditions, 

a-[- X a— 4b 

— - — = 6 + a: ; whence, x = — — — . 

Suppose a = 54, and b =:9; then x = =r — -= 6. 

o o 

The father being 54 years old, and the son 9, in 6 year^ the 
fawher will be 60 years old, and his son 15 ; now 15 is the 
ff urth of 60 ; hence, a; = 6 satisfies the enunciation. ^ t 

Let us now suppose a = 45, and b = 15 ; 

45 .._ BU 
then, V = = — 5, 



108 ELEMENTS OF ALGEBKA. [CH^iP. IV. 

If we substitute this value of x iii the equation, 

a-\-x 

-4- = ^ + ^, 

45-5 ,^ ^ 
we obtam, = lo — 5 ; 

or, 10 = 10. 

Hence, — 5 substituted for x^ verifies the equation, and ther&- 
fore is a true answer. 

Now, the positiv^e result which was obtained, shows that the 
age of the father will be four times that of the son at the 
expiration of 6 years from the time w^hen their ages were 
considered ; while the negative result, indicates that the age of 
the father was four times that of his son, 5 years 'previous to 
the time when their ages were compared. 

The question, taken in its general, or algebraic sense, demands 
the time, when the age of the father is four times that of the 
son. In stating it, we supposed that the time was yet to 
come ; and so it was by the first supposition. But the con- 
ditions imposed by the second supposition, required that the 
time should have already passed, and the algebraic result con-^ 
formed to this condition, by appearing with a negative sign. 

Had we wished the result, under the second supposition, to 
have a positive sign, we might have altered the enunciation 
by demanding, how many years since the age of the father loas 
four times that of the son. 

If X denote the number of years, we shall have from the 

conditions, 

a — X ^ . 46 — a 

— - — = — X : hence, x = — - — . 

4 '3 

If a = 4:5 and b = 15, x wdll be equal to 5. 

From a careful consideration of the preceding discussion, we 

may deduce the following principles with regard to negative 

restilts. 

1st. Every negative value found for the unknown qumitiiy from 
an eqxLation of the first degree, will, ivhen taken with ^H proper 
sign, satisfy the equation from tohich it was derived. 



CHAP. lY.] EQUATIONS OF THE FIRST DEGREE. 109 

2d. This negative value, taken with its proper sigu, vnll also 
satisfi the conditions of the problem, un-lerstood in its algebraic 
sense. 

3d. If a positive result is interpreted in a certain sense, a nega- 
tive result must be interpreted in a directly contrary sense. 

4th. The negative result, with its sign changed, may be regarded 
€LS the ansiver to a problem of which the enunciation only differs 
from the one proposed in this : that certain quantities which were 
additive have become subtractive, and the reverse. 

90« As a further illustration of the extent and power of the 
algebraic language, let us resume the general problem of the 
laborer, already considered. 

Under the supposition that the laborer receives a sum c, we 

have the equations 

X -^ y ^^n) . bn -\- c an — c 

Y whence, x — -, y = —z-. 

ax — by = c ) a -{- b a -{- b 

If, at the end of the time, the laborer, instead of receiving 
a sum c, owed for his board a sum equal to c, then, by would 
be greater than ax, and under this supposition, we should have 
the equations, 

X -\- y = n, and ax — by z=z — c. 
Now, since the last two equations differ from the preceding 
two given equations only in the sign of c, if we change the 
sign of c, in the values of x and y, found from these equations, 
the results will be the values of x and y, in the last equa- 
tions : this gives 

bn — c an -\- c ■ 

"" "= ^+~6' ^ ^ ~V+b' 
The results, for both enunciations, may be comprehended in 
the same formulas, by writing 

_ 6;i ± c _ an zp c 

^ ~ a+~6' ^ "^ ~V+b' 

The double sign zfc, is read plus or minus, and q=, is read, 
minus or plus. The upper signs correspond to the case in 
which the laborer received, and the lower signs, to the case in 



110 ELEMENTS OF ALGEBRA. [CHAP. IV. 

which he owed a sum c. These formulas also comprehend the 
case in \Yliich, in a settlement between the laborer and hid 
employer, their accounts balance. Tliis supposes c = 0, which 
gives 

hn an 



a + 6' ^~a + 6' 

Discussion of Problems. 

91. The discussion of a problem consists in making every 
possible supposition upon the arbitrary quantities which enter 
the equation of the problem, and interpreting ine results. 

An arbitrary quantity, is one to which we may assign a value^ 
at pleasure. 

In every general problem there is always one or more arbi 
trary quantities, and it is by assigning particular values to 
these that we get the particular cases of the general problem. 

The discussion of the following problem presents nearly all 
the circumstances which are met with in problems giving rise 
to equations of the first degree. 

PROBLEM or THE COURIERS. 

Two couriers are traveling along the same right line and 
in the same direction from W toward R. The number of miles 
traveled by one of them per hour is expressed by m, and the 
number of miles traveled by the other per hour, is expressed 
by n. Now, at a given time, say 12 o'clock, the distance be- 
tween them is equal to a number of miles expressed by a : re- 
quired the time when they are together. 

W A B R^ 

At 12 o'clock, suppose the forward courier to be at B, the 
other at A, and R or R' to be the point at which they are 
together. 

Let a denote the distance AB, between the couriers at 12 
o'clock, and suppose that distances measured to the right, froin 
A, are regarded as positive quantities. 



CHAP. IV. J EQUATIONS OF THE FIRST DEGEEE. Ill 

Let t denote the number of hours frcm 12 o'clock to the 
time when they are together. 

Let X denote the distance traveled by the forward courier 
in t hours ; 

Then, a-\- x will denote the distance traveled by the other 
in the same time. 

Now, since the rate per hour, multiplied by the number of 
hours, gives the distance passed over by each, we have, 
t xm=:a-\- X - - ' - (1) 
txn—x - . - . (2). 

Subtracting the second equation from the first, member from 

member, we have, 

t(m — w) = a ; 

a 

whence, - - - - ^ = . 

m — 7h 

"We will now discuss the value of ^ ; a, m and n, being 

arbitrary quantities. 

First^ let us suppose m y n. 

The denominator in the value of t^ is then positive, and since 
a is a positive quantity, the value of t is also positive. 

This result is interpreted as indicating that the time when 
they are together is after 12 o'clock. 

The conditions of the problem confirm this Interpretation. 

For if m > n^ the courier from A will travel faster than the 
courier from B, and will therefore be continually gaining on 
him : the interval which separates them will diminish more and 
more, until it becomes 0, and then the couriers will be found 
upon the same point of the line. 

In this case, the time t^ which elapses, must be added to 12 
o'clock, to obtain the time when they are together. 

Second^ suppose m < n. 

The denominator, in — n will then be negative, and, the value 
of t will also be negative. 

Tlii« result is interpreted in a sense exactly contrary to the 
interpretation of the positive result ; that is, it indicates that 
the time of their being together was previous to 12 o'clock. 



/ 



112 ELEMENTS OF ALGEBRA. [CHAP. IV. 

This interpretation is also confirmed by considering tlie 
circumstances of the problem. For, under the second suppo- 
sition, the courier which is in advance travels the fastest, and 
therefore vtdll continue to separate himself from the other 
courier. At 12 o'clock the distance between them was equal 
to a : after 12 o'clock it is greater than a ; and as the rate 
of travel has not been changed, it follows that previous to 12 
o'clock the distance must have been less than a. At a certain 
hour, therefore, before 12, the distance between them must have 
been equal to nothing, or the couriers were together at some 
point E'. The precise hour is found by subtracting the value of 
t from 12 o'clock. 

Third, suppose m = n. 

The denominator m — n will then become 0, and the value 

of t will reduce to -, or oo . 

This result indicates that the length of time that must elapse 
before they are together is greater than any assignable time, or 
in other words, that they will never be together. 

This interpretation is also confirmed by the conditions of' the 
p-oblem. 

For, at 12 o'clock they are separated by a distance a, and if 
m = n they must travel at the same rate, and we see, at once, 
that whatever time we allow, they can never come together j 
hence, the time that must elapse is infinite. 

Fourth, suppose a z=: and m > ?i or m <^ n. 

The numerator being 0, the value of the fraction is oi 

This result indicates that they are together at 12 o'clock, 
or that there is no time to be added to or subtracted fi'om 
12 o'clock. 

The conditions of the problem confirm this interpretation. 
Because, if a = 0, the couriers are together at 12 o'clock ; and 
since they travel at different rates, they could never have been 
together, nor can they be together after 12 o'clock: hence, t car 
have no other value than 0. 



«^BA.P. IV. J OF INEQUALITIES. US 

Fiflhj suppose a = and m = 71. 
The value of t becomes -, an indeterminate result. 

This indicates that i may have any value whatever, or in 
other words, that the couriers are together at any time e'.thei 
before or after 12 o'clock: a«nd this too is evident from the cir 
cumstances of the problem. 

For, if a = 0, the couriers are together at 12 o'clock ; and 
since they travel at the same rate, they will always be together; 
hence, t ought to be indeterminate. 

The distances traveled by the couriers in the time . are, 

respectively, ^ 

ma _ na 

and 



m — 7L m — n 

both of which will be plus when m > n, both miTius when m <^n, 
and irifinite when m =1 n. ^ 

In the first case t is positive ; in the second, negative ; and in 
the third, infinite. 

When the couriers are together before 12 o'clock, the distances 
are negative, as they should be, since we have agreed to call 
distances estimated to the right positive^ and from the rule for 
interpreting negative results, distances to the left ought to be 
..'egarded as negative. 

Of Inequalities. 

92. An inequality is the expression of two unequal quantities 
connected by the sign of inequality. 

Thus, a > & is an inequality, expressing that the quantity a 
is greater than the quantity h. 

The part on the left of the sign of inequality is called the firsi 
member^ that on the right the second member. 

The operations which may be performed upon equations, may 
in general be performed upon inequalities; but there are, never- 
theless, some exceptions. 

In order to be clearly understood, we will give exam})les 0/ 
the difi'ereut transformations to which ir equalities may be suli 

8 



114 ELEMENTS OF ALGEBRA [CHAP. IV, 

jected. taking care to point out the exceptions to which these 
transfcrmations are liable. 

Two inequalities are said to subsist in the same sense, when 
Ihe greater quantity is in the first member in both, or in the 
second inember in both: and in a contrary sense, when ihe 
greater quantity is in the first member of one and in the second 
inember of the other. 

Thus, 25 > 20 and 1 8 > 10, or 6 < 8 and 7 < 9, 
are inequalities which subsist in the same sense ; and the in 
equalities 

15 > 13 and 12 < 14, 
subsist in a contrary sense. 

1. If loe add the same quantity to both members of an inequality ^ 
or subtract the same quantity from both members, the resulting 
inequality will subsist in the same sense. 

Thus, take 8 > 6 ; by adding 5, we still have 
o-r-5>6 + 5; 
and subtracting 5, we liaA'-e 

8 - 5 > 6 - 5. 

When the two members of an inequality are both negative, 
that one is the least, algebraically considered, which contains the 
greatest number of units. 

Thus, — 25 < — 20 ; and if 30 be added to both members, 
we have 5 < 10. This must be understood entirely in an alge- 
braic sense, and arises from the convention before established, to 
<ionsider all quantities preceded by the minus sign, as subtractive. 

The principle first enunciated serves to transpose certain terriis 
from one member of the inequality to the other. Take, for ex 
ample, the inequality 

a2 + 62>362_2a2; 

there will result, by transposing, 

a2 + 2a2 > 3^2 - 52^ or 3a2 > 262. 

2. If two inequalities subsist in the same sense, and we add them 
member to member, the resulting inequality will also subsist ifi the 
same sense. 



CHAP. IV.J OF INEQUALITIES. 115 

Thus, if we add a > 6 and c"^ d, member to member, 
there results a + c > 5 + c?. 

But this is not ahoays the case, when we subtract, member from 
member, two inequalities established in the same sense. 

Let there be two inequalities 4 < 7 and 2 < 3, we have 

4 - 2 or 2 < 7 - 3 or 4. 
But if we have the inequalities 9 < 10 and 6 < 8, by sub- " 
ti acting, we have 

9—6 or 3 > 10 ~ 8 or 2. 

We should then avoid this transformation as much as possible, 
or if we employ it, determine in which sense the resulting in- 
equality subsists. 

3. If the two members of an inequality be multiplied by a 
positive quantity, the resulting inequality will exist in the same 
sense. 

Thus„ - - - a <C. b, will give 3a < 35 ; 
and, - - - - — a < — 5, — 3a < — 36. 

This principle serves to make the denominators disappear. 

r/2, 7)2 /j2 _^ f~J2 

Erom the inequality — — - — > — , 

we deduce, by multiplying by 6ad, 

3a(a2 -62) y2d{c'^ -d^), 
and the same principle is true for division. But, 

When the two members of an inequality are multiplied or 
divided by a negative quantity^ the resulting inequality will sub- 
tist in a contrary sense. 

Take, for example, 8 > 7 ; multiplying by — ■ 3, we have 
-24< -21. 

Q 8 7 

In like manner, 8 > 7 gives — —, or ^ < 5~« 

— o o o 

Therefore, when the two members of an inequality are multi- 
plied or divided by a quantity, it is necessary to ascertain 
whether the multiplier or ^ivisor is negative; for, in that case, 
the inequality will exist in a contrary sense. // 



116 ELEMENTS OF ALGEBRA. [CHAP. IV 

4. It is v,oi permitted to change the signs of the two members 
^ an inequality^ unless we establish the resulting inequality in a 
contrary sense; for, -this transformation is evidently the same as 
eon] tiplying the two members by — 1. 

5. Both members of an inequality between positive quaniikes 
ean be squared, and the inequality will exist in the same sense. 

Thus, from 5 > 3, we deduce, 25 > 9 ; from a + 6 > c, we 

find 

(a + by > c\ 

6. When the signs of both members of the inequality are not 
known, we cannot tell before the operation is performed, in which 
sense the resulting inequality will exist. 

Por example, — 2 < 3 gives ( — • 2)^ or 4 < 9. 

Butj 3 > — 5 gives, on the contrary, (S)^ or 9 < ( — 5)^ 
or 25. 

We must, then, before squaring, ascertain the signs of the two 
members. # 

Let us apply these principles to the solution of the following 
examples. By the solution of an inequality is meant the oper 
ation of finding an in,equality, one member of which is the 
imknown quantity, and the other a known expression. 



Ans. X > 5. 
Ans. ar ^ 4. 



-f- 6a; — a5 > -— Ans. « > a. 





EXAMPLES. 


1. 


5a: - 6 > 19. 


2. 


3a; + 11 a;- 30 > 10. 



5 ' "5 

y - oa; -i- a6 < y 



— ax '\- ah <^ — -. Ans, x <^b. 



CHAPTER V. 

EXTRACTION OP THE SQUARE ROOT OF NUMBERS.- -FORMATION OF TH» 
• SQUARE AND EXTRACTION OF THE SQUARE ROOT OF ALGEBRAIC QUANTI- 
TIES. TRANSFORMATION OF RADICALS OF THE SECOND DEGREE. 

93 • The square or second power of a number, is the product 
which arises from multiplying that number by itself once : foT 
example, 49 is the square of 7, and 144 is the square of 12. 

The square root of a number, is that number which multiplied 
by itself once will produce the given number. Thus, 7 is the 
square root of 49, and 12 the square root of 144 : for, 7x7 = 49, 
and 12 X 12 = 144. 

The square of a number, either entire or fractional, is easily 
found, being always obtained by multiplying the number by itself 
once. The extraction of the square ro§t is, however, attended 
with some difficulty, and requires particular explanation. 

The first ten numbers are, 

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
and their squares, 

1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 

Conversely, the numbers in the first line, are the square roots 
of the corresponding numbers in the second line". 

We see that the square of any number, expressed by one 
figure, will contain no unit of a higher order than tens. 

The numbers in the second line are jperfeci squares, and, 
generally, any number which results from multiplying a whole 
number by itself once, is a perfect square. 

If we wish to find the square root of any number less thaa 
100, we look in the second line, above given, and if the nunir 
ber is there written, the corresponding number in the first line 



118 ELEMENTS OP ALGEBEA. TCHAP. V. 

is its square root. If the number falls between any two num 
bers in the seo-ond line, its square root will fall between the 
corresponding numbers in the first line. Thus, 55 falls between 
49 and 64 ; hence, its square root is greater than 7 and less 
than 8. Also, 91 falls between 81 and 100; hence, its square 
root is gi'eater than 9 and less than 10. 

If now, we change the units of the first line, 1, 2, 3, 4, &c., 
into units of the second order, or tens, by annexing to each, 
we shall have, 

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 
imd their corresponding squares will be, 

100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000: 
Hence, the square of any number of tens will contain no unit of 
« less denomination than hundreds. 

94. We may regard every number as composed of the sun? 
of its tens and units. 

JN[ow, if we represent any number by N, and denote the 
tens by o, and the units by b, we shall have, 

whence, by squaring both members, 

i\J2 ^ a2 4. 2ab + 52 : 

Hence, the square of a number is equal to the square of the 
(ens, plus twice the product of the tens by the units, j^lus the squure 
of the units. 

For example, 78 = 70 + 8, hence, 

(78)2 =: (70)2 + 2 X 70 X 8 + (8)2 ^ 4900 + 1120 + 64 = 6084. 

95. Let us now find the square root of 6084. 
Since tliis number is expressed by more than two 

figures, its root will be expressed by more than one. 60 84 

But since it is less than 10000, wliich is the square 
of 100, the root will contain but two pkces of figures ; that 
is, i:mts and tens. 

Now, the square of the number of tens must be found in the 
number expressed by the two left-hand figures, which we will 
separate from the other two, by placing a point over the place 



60 84 1 78 
49 
7 X 2 = 148 







CHAP, v.] SQUARE ROOT OF NUMBERS. 119 

of units, and another over the place of hundreds. These parts, 
of two figures each, are called periods. The part 60 is com- 
prised between the two squares 49 and 64, of which the roots 
are 7 and 8 : hence, 7 is the number of tens sought ; and the 
required root is composed of 7 tens plus a certain number of 
units. 

The number 7 being found, we 
set it on the right of the given 
number, from which we separate 

it by a vertical line: then we 7x2 = 14 8 118 4 

subtract its square 49 from 60, 118 4 

which leaves a remainder of 11, 
to which we bring down the two 
next figures 84. The result of this operation is 1184, and this 
number is made up of twice the product of the tens by the units 
plus the square of the units. 

But since tens multiplied by units cannot give a product of a 
lower order than tens, it follows that the last number 4 can 
form no part of double the product of the tens by the units : 
this double product, is, therefore found in the part 118. 

Now, if we double the nvimber of fens, VY^hich gives 14, and 
then divide 118 by 14, th . quotient 8 is the number of units of 
the root., or a greater number. This quotient can never be too 
small, since the part 118 will be at least equal to twice the 
product of the number of tens by the units: but it may be too 
large; for the 118, besides the double product of the number 
of tens by the units, may likewise contain tens arising from 
the square of the units. 

To ascertain if the quotient 8 expresses the number of units, 
we place the 8 to the right of the 14, which gives 148, and then 
we multiply 148 by 8 : Thus, we evidently form, 

1st, the square of the units ; and 

2d, the double product of the .ens by the units. 

This multiplication being affected, gives for a product 1184, 
the same number as the result of the first operation. Having 



120 ELEMENTS OF ALGEBRA. [CHAP. V, 

subtracted the product, we find the remainder equal to : hence 
78, is the root required. 

Indeed, in the .operations, we have merely subti acted from the 
given number 6084, 

1st, the square of 7 tens or of 70 ; 

2d, twice the product of 70 hy 8; and 

3d, the square of 8 : that is, the three parts which enter mto 
tiie composition of the square of 78. 

In the same manner we may extract the square root of any 
number expressed by four figures. 

95. Let us now extract the square root of a number expressed 
by more than four figures. 

Let 56821444 be the number. 56 82 14 44 | 7538 

If we consider the root " as the 49 

sum of a certain number of tens 14 5 
and a certain number of units, the 
given number will, as before, be 150 3 

equal to the sauare of the tens plus 
twice the product of the tens by 
the units plus the square of the units. 
If then, as before, we point off 
a period of two figures, at the right, the square of the tens of the 
required root will be found in the number 568214, at the left ; 
and the square root of the greatest perfect square in this number 
will express the tens of the root. 

But since this number, 568214, contains more than two figures, 
its root will contain more than one, (or hundred's), and the sqaare 
of the hundreds will be found in the figures 5682, at the left of 14 ; 
hence, if we poi»it off a second period 14, the square root of the 
greatest perfect square in 5682 will be the hundreds of the required 
root. But since 5682 contains more than two figures, its root will 
contain more than one, (or thousands), and the square of the thousands 
will be found in 56, at the left of 82 : hence, if we point off a third 
period 82, the square root of the greatest perfect square in 56 will 
be the thousands of the required root. Hence, we place a point 
over 56. and then nroceed thus : 



78 2 

72 5 
57i4 
450 9 



1506 8 



12054 4 
12054 4 



(!HAP. V.J SQUARE ROOT OF NUMBERS. 121 

Placing 7 on the right of the given number, and subtracting 
its square, 49, from the left hand period, we find 7 for a remain- 
der, to which we annex the next period, 82. Sepai'ating the last 
figure at the right from the others by a point, and dividing the 
number at the left by twice 7, or 14, we have 5 for a quotient 
figure, which we place at the right of the figure already found, 
and also annex it to 14. Multiplying 145 by 5, and subtracting 
the product from 782, we find the remainder 57. Hence, 75 is 
the number of tens of tens, or hundreds, of the required square 
root. 

To find the number of tens, bring down the next period and 
annex it to the second remainder, giving 5714, and divide 571 
by double 75, or by 150. The quotient 3 annexed to 75 gives 
753 for the number of tens in the root sought. 

We may, as before, find the number of units, which in this 
case will be 8. Therefore, the required square root is 7538. A 
similar course of reasoning may be applied to a number expressed 
by any number of figures. Hence, for the extraction of the 
square root of numbers, we have the following 

RULE. 

I. Separate the given number into 'periods of iivo figures each, 
beginning at the right hand: the period on the left will often con- 
tain but one figure. 

II. Find the greatest perfect square in the first period on the 
left, and p>lace its root on the right after the manner of a quotient, 
in division. Subtract the square of this root from the first 
■period, and to the remainder bring down the second period for a 
dividend. 

III. Double the root already found and place it on the left for a 
divisor. See how many times the divisor is contained in the 
dividend^ exclusive of the right hand figure, and place the quotient 
i?i the root and also at the right of the divisor. 

IV. Multiply the divisor, thus augmented, by the last figure 
of the root found, and subtract the product from the divideiid 



J 22 ELEMENTS OF ALGEBRA. CHAP. V. 

and to the remai'.nder bring down the next period for a new 
dividend. 

V. Double the whole root already found^ for a new divisor, 
and continue the operation as before, until all the periods are 
brought down. 

Remark I. — If, after all the periods are brought down, there. 
is no remainder, the proposed number is a perfect square. But 
if there is a remainder, we have only found the root of the 
greatest perfect square contained in the given number, or the 
entire part of the root sought. 

For example, if it were required to extract the square root of 
168, we should find 12 for the entire part of the root and a 
remainder of 24, which shows that 168 is not a perfect square. 
But is the square of 12 the greatest perfect square contained 
in 168 ? That is, is 12 the entire part of the root 1 

To prove this, we will first show that, the difference between 
the squares of two consecutive numbers, is equal to twice the less 
number augmented by 1. 

Let a represent the less number, 

and a 4- 1, the greater. , 

Then, (a + 1)^ = a2 j^^a+l, 
and (a)2 = a^, 



their difference is 2a + 1 as enunciated : hence, 

The entire part of the root cannot be augmented by 1, unless 
the remainder is equal to, or exceeds twice the 'root found, plus 1. 

But, 12 X 2 4- 1 = 25 ; and since the remainder 24 is less 
than 25, it follows that 12 cannot be augmented by a number 
as great as unity : hence, it is the entire part of the root. 

The principle demonstrated above, may be readily applied in 
finding the squares of consecutive numbers. 

If the numbers are large, it will be much easier to apply the 
a])0ve pifnciple than to square the numbers separately. 



CHAP, v.] SQUARE ROOT OF NUMBERS. 12B 

For example, if we have (651)2 ^ 423801, 
and wish to find the square of 652, we have, 
(651)2 = 423801 
+ 2 X 651 =: 1302 

+ 1 .= 1 

and (652)2 =z 425104. 

Also, (652)2 ^ 425104 

+ 2 X 652 = 1304 

+ 1 -__ 1 

and (653)2 ^ 426409. 

Remark IL — The number of places of figures in the root 
will always be equal to the number of periods into which the 
given number is separated. 

EXAMPLES. 

1. Find the square root of 7225. 

2. Find the square root of 17689. 
3.* Find the square root of 994009. 

4. Find the square root of 85678973. 

5. Find the square root of 67812675. 

6. Find the square root of 2792401. 

7. Find the square root of 37496042. ^ 

8. Find the square root of 3661097049. 

9. Find the square root of 918741672704. 

Remark III. — The square root of an imperfect square, is in 
commensurable with 1, that is, its value cannot be expressed 
in exact parts of 1. 

To prove this, we shall first show that if -r- is an irreduci- 

b 

a2 
ble fraction, its square — ^ must also be an irreducible fraction. 

A number is said to be prime when it cannot be exactly di- 
vided by any other number, except 1. Thus 3, 5 and 7 are 
prime number u 



124 ELEMENTS OF ALGEBEA. [CHAP. V. 

It is a fundamental principle,- that every number may be re- 
solved into prime factors, and that any number thus resolved, 
is equal to the continued product of all its prime factors. It 
often happens that some of these factors are equal to each 
other. For example, the number 

50 = 2 X 5 X 5 ; and, 180 =: 2 X 2 X 3 X 3 X 5. 

Now, from the rules for multiplication, it is evident that the 
square of any number is equal to the continued product of all 
the prime factors of that number, each taken twice. Hence, we 
see that, the square of a nuinher cannot contain any prime factoi 
which is not contained in the number itself. 

But, since — -, is, by hypothesis, an irreducible fraction, a 

and b can have no common factor : hence, it follows, from 
what has just been shown, that a^ and b'^ cannot have a com- 

mon factor, that is, -— is an irreducible fraction, which was 

0^ 

to be proved. 

For like reasons, — -, — -, - - -— , are also irreducible fractions. 

Now, let c represent any whole number which is an imper 
feet square. If the square root of c can be expressed by a 
fraction, we shall have 






in which — - is an iiTcducible fraction. 



Squaring both members, gives. 



^ = !5-' 



or a whole number equal to an irreducible fraction, which is 
absurd ; hence, Vc" cannot be expressed by a fraction. 

We conclude, therefore, that the square root of an imperfect 
square cannot be expressed in exact parts jf 1. It may be 
shown, in a similar manner, that any root of an imperfect 
power of the degree indicated, cannot he expressed in exact parts 
of 1. 



CHAP. Vj SQUARE ROOT OF FRACTIONS. 125 

Extraction of the Square Root of Fractions. 

96 » Since the second power of a fraction is obtained by- 
squaring the numerator and denominator separately, it follows 
that the square root of a fraction will be equal to the square root 
of the numerator divided by the square root of the denominator. 



For example, 



fofi a 

V T2 "" y 



a a a? 

'""'^ , T ^ T = 6^- 

But if the numerator and the denominator are not both per- 
fect squares, the root of the fraction cannot be exactly found. 
We can, however, easily find the root to within less than the 
fractional unit. 

Thus, if we were required to extract the square root of the 

fraction — , to within less than — , multiply both terms of the 

fractions by b, and we have — . 

Let r^ represent the greatest perfect square in ab, then will 
a^be contained between r^ and (?• + 1)^, and — will be con- 
tained between 

and the true square root of r^r = -;-, will be contained be- 

0^ 

fcween 

r r 4- 1 

T ""-^ -J-' 

r r -{- I 1 

but the difference between -— and — z — is — -: hence, either 

o b b 

will be the square root of — , to within less than — . We have 



then the following 



126 ELEMENTS OF ALGEBRA. ICHAP. V, 

RULE. 

Multiply the numerator by the denominator, and extract tht 
square root of the product to within less than 1 ; divide the 
result by the denominator, and the quotient will be the approxi- 
mate root. 

• g 

For example, to extract the square root of -— -, we multiply 

o 

3 by 5, which gives 15 ; the perfect square nearest 15, is 16, 

and its square root is 4 ; hence, -^ is the square root of -— 

5 o 

to mthin less than -— . 
5 

97« If we wish to determine the square root of a whole 

number which is an imperfect square, to within less than a 

giv^n fractional unit, as — , for example, we have only to place 

the number under a fractional form, having the given fractional 
unit (Art. 63), and then we may apply the preceding rule : or 
what is "an equivalent operation, we may 

Multiply the given number by the square of the denominator 
of the fraction which determines the degree of approximation ; then 
extract the square root of the product to the nearest unit, and 
divide this root by the denominator of the fraction. 

EXAMPLES. 

1. Let it be required to extract the square root of 59, to 
within less than — . 

First, (12)2 = 144; and 144 X 59 = 8496. 

Now, the square root of 8496 to the nearest unit, is 92 : henc6 

92 1 

YH — '^T2^ which is true to within less than — -. 
1<<J " 12 



2. Find the -/iT to within less than -— . Jns'. 3 A- 

3. Find the ^223 to within less than — . Ans. 14f|. 



CHAP, v.] SQUARE ROOT OF FRACTIONS. 127 . 

97*» The manner of determining the approximate root in deci- 
mals, is a consequence of the preceding rule. 

To obtain the square root of an entire number within —■, 

fnn' innn ' ^^'^ ^^ ^^ ^^^^ necessary, according to the preceding 
rule, to multiply the proposed number by (lO)^, (lOO)^, (1000)2; 
or, which is the same thing, 

Annex to the number, two, four, six, <&€., ciphers: then extract 
the root of the product to the nearest unit, and divide this root 
hy, 10, 100, 1000,- &c., luhich is effected hg pointing off one, two^ 
three, dtc, decimal places from the right hand. 

EXAMPLES. 

1. To find the square root of 7 to within less than — — . 

Having multiplied by (lOO)^, that is, 
having annexed four ciphers to the right 
hand of 7, it becomes 70000, whose . 
root extracted to the nearest unit, is 264, 
which being divided by 100 gives 2.64 
for the answer, and this is true to within 

less than j^. "^ ^^^^ 

2. Find the .^29 to within less than -— . Ans. 5.38. 

3. Find the ^/227 to mthin less than — — -. Ans. 15.0665. 



7 00 00 

4 

300 

276 



2:04 



2400 
2096 



10000 

Remark. — The number of ciphers to be annexed to the whole 
number, is always double the number of decimal places required 
to be found in the root. 

98. The manner of extracting the square root of a number 
containing an entire part and decimals, is deduced imm.ediately 
from the preceding article. 

Let us take for example the number 3.425. This s equiva- 

3425 

lent to . Now, 1000 is not a perfect square, but 'the de^ 



128 ELEMENTS OF ALGEBRA. [CHAP. V. 

nominator may be made such without altering the value of the 
fraction, by multiplying both terms by 10 ; this gives 

34250 ^ 34250 

10000 ^'' (100)2- 
Then, extracting the square root of 34250 to the nearest unit, 

we find 185 ; hence, — — or 1.85 is the required root to with- 
in less than — — -. 

If greater exactness be required, it will be necessary to annex 
to the number 3.425 as many ciphers as shall make the num- 
ber of periods of decimals equal to the number of decimal 
places to be found in the root. Hence, to extract the square 
root of a mxixed decimal : 

Annex ciphers to the proposed number until the whole number 
of decimal places shall be equal to double the number required in 
the root. Then^ extract the root to the nearest unit^ and point off^ 
from, the right hand^ the required number of decimal places. 

EXAMPLES. 



1. Find the ^327 1.4707 to within less than .01. 

Ans. 57.19. 



2. Find the y/ 31.027 to within less than .01. Ans. 5.57. 

3. Find the ^ 0.01001 to within less than .00001. 

Ans. 0.10004. 

99. Finally, if it be required to find the square root of a 
vulgar fraction in terms of decimals : 

Change the vulgar fraction into a decimal and continue the di- 
vision until the number of decimal places is double the number 
required in the root. Then, extract the root of the decimal by the 
last rule. 

EXAMPLES. 

1. Extract the square root of — to witliin less than .001 

14 

This number, reduced to decimals, is 0.785714 to within less 
than 0.000001. The root of 0.785714, to the nearest unit, is 



CHAP, v.] SQUARE ROOT OF ALGEBRAIC t^UANTITIES. 129 

886 : hence, 0.88G is the root of — - to within less than .001. 
2. Find the y^2i| to within less than 0.0001. ^ns. 1.6931. 

Extraction of the Square Root of Algebraic Quantities. 

100. Let us first consider the case of a monomial. 

In order to discover the process for extracting the square 
root, let us see how the square of a monomial is formed. 

By the rule for the multiplication of monomials (Art. 42), 
we have 

(5a%hy =: 5a^3c X 5a'^h'^c = 25a^b^c^ ; 
that is, in order to square a monomial, it is necessary to 
square its co-efficient, and double the exponent of each letter. 

Hence, to find the square root of a monomial, 

Extract the square root of the co-efficient for a neiu co-efficient, 
and write after this, each letter, with an exponent equal to its 
fviginal exponent divided by two. 

Thus, y64a66* =. %aW ; for, ^aW X ^aW = Q4:a^b\ 
and, y &25d'b^c^ = 2bab^c^ ; for, (25ab^c^Y = 625a^bh\ 

From the preceding rule, it follows, that, when a monomial 
is a perfect square, its numerical co-efficient is a perfect square, 
and every expoyient an even number. 

Thus, 25a*62 ^g ^ perfect square, but 98a6'* is not a perfect 
•square ; for, 98 is not a perfect square, and a is affected witli 
jn uneven exponent. 

Of Polynomials. 

101. Let us next consider the case of polynomials. 

Let K denote any polynomial whatever, arranged with refer- 
etice to a certain letter. Now the square of a polynomial is 
the product arising from m\iltiplying the polynomial by itself 
©nee : hence, the first term of the product, arranged with refer- 
ence to a particular letter, is the square of the first term of 
the polynomial, arranged witn reference to the same letter. 

9 



130 ELEMENTS OF ALGEBRA. [CHAP. V. 

Therefore, the square root of the first term of such a product 
will be the first term of the required root. 

Denote this term^ by r, and the followig terms of the root, 
arranged with reference to the leading letter of the polynomial, 
by r\ r", r'", &c., and we shall have 

iV= (r + r' + r" -h r'" + &c.;)2 
or, if we designate the sum of all the terms of the root, after 
the first, by 5, 

N={r-{-sY = r'^ + 2rs + s^ 

rr: r2 + 2r (r' 4- r" + r'" + &c.) + s\ 
If now we subtract r^ from iV, and designate the remainder 
by M, we shall have, 

I^-r^ = Ji = 2r{r'-{- r" + r'" + &c.) + s\ 
which remainder will evidently be arranged with reference to 
the leading letter of the given polynomial. If the indicated 
operations be performed, the first term 2rr' will contain a 
higher power of the leading letter than either of the following 
terms, and cannot be reduced with any of them. Hence, 

If the first term of the first remainder he divided hy twice the 
first term of the root, the quotient will be the second term of 
the root. 

If now, we place r -\- r' =n, 

and designate the sum of the remaining terms of the root, 
r'\ r"\ &;c., by s\ we shall have 

N—{n-\- s')2 =71^ -\- 2ns' + s'\ 
If now we subtract n^ from iV", and denote the remainder 
by B\ we shall have, 

N-n?=^B' = 2ns' + s'^ = 2{r + r') {r" + r'" + &c.) -f s'>; 
in which, if we perform the multiplications indicated in the 
second member, the term 2rr" will contain a higher power of 
the leading letter than either of the following terms, and can- 
not, consequently, be reduced with any of them. Hence, 

Jf the first term of the second remainder be divided hy twic* 
the first term of the root, the quotient will be the third term 
of the root. 



CHAP, v.] SQUARE EOOT OF ALGEBRAIC QUANTITIES. Ul 

If we make 

r-\-r' -\-r" = n\ and r'" -|- r'^ + &;c. = s'\ 
we shall have 



N= (ii' 4-5")' = ^'^ + 272 '5" + s"2 . and 



^ 



N-n"' = R' =r 2 (r + r' + r") {r'" + r^ -h &c.) + s" ^ *, 
in which, if we perform the operations indicated, the terrpi 
2rr'" will contain a higher power of the leading letter than 
any following term. Hence, 

If we divide the first term of the third remainder hy twice 
the first term of the root, the quotient will be the fourth terra 
of the root. 

If we continue the operation, we shall see, generally, that 

The first term of any remainder^ divided hy twice the first 
term of the root, will give a new term of the required root. 

It should be observed, that instead of subtracting ti^ from 
the given polynomial, in order to find the second remainder, 
that that remainder could be found by subtracting (2r -|- r')r' 
from the first remainder. So, the third remainder may be found 
by subtracting ^n + r")r" from the second, and similarly for 
the remainders which follow. 

Hence, for the extraction of the square root of a polynomial, 
we have the following 

RULE. 

I. Arrange the polynomial with reference to one of its letters j 
rind then extract the square root of the first term, which will give 
the first term of the root. Subtract the square of this term from, 
the given polynomial. 

II. Divide the first term of the remainder by twice the first term 
of the root, and the quotient loill be the second term of the root. 

III. From the firs*, remainder subtract the product of twice the 
first term of the root plus the second term, by the second term. 

IV. Divide the first term of the second remainder by twice the 
first term of the root, and the quotient will be the third term of 
the root. 



1S2 ELEMENTS OF ALGEBRA. ^CHAP. V, 

V". From the second remainder subtract the product of twice the 
sum of the first and second terms of the root, plus the third 
term, hy the third term, and the result will he the third t^emain- 
der,J^rom which the fourth term of the root may he found as 
lefore. 

VI. Continue the operation till a remainder is found equal to 
0, or till the first term of some remainder is not divisible hy 
twice the first term of the root. In the former case the root found 
is exact, and the polynomial is a perfect square ; in the latter 
dose, it is an imperfect square. 

EXAMPLES. 

1. Extract the square root of the polvnomial 

49a252 — 24a63 -f 25 a* — SOa^^, -\- 165*. 

First arrange it with reference to the letter a. 



25a* — 30a35 + 49a252 — 24.aP + 166' 
25a* 



R = 


- 30a35 + 49a252 _ 
-30a36+ 9a262 


-24a63+ 166* 


W = 


+ 40a262 - 
+ 40a262 _ 


- 24a63 4- 166* 

- 24a63 -f 166* 


R"=i 







5a2 - 3a6 + 462 



10a2- 


-3a6 
-3a6 


- SOa^^ftf- 9a262 


10a2 


- 6a6 + 462 
46^- 



40a262 - 24a63 + 166*. 



2. Find the square root of 

a* 4- 4a3.r -{- Qa'^x^ ■{- Aax'^ -|- a;*. 

8. Find the square root of 

a* — 2a^x -f 3a2a;2 - 2a;r3 + xK 

4. Find the square root of 

4r6 -f- I2a;5 + 5a:* - 2a;3 + Ix'^ - 2a; + 1. 

5. Find the square root of 

92* - 12a36 + 28a262 - 16a63 -f 166*. 

6. Find the square root of 

^5a4i2 _ 40a362c 4- 76a262c2 — 48a62c3 + 3662c* — 30a*6c f 240^6^ 
— 36a26c3 4- 9a*c2. 



CHAP, v.] RADICALS OF THE SECOND DEGREE. 133 

Remai'Jcs on the Extraction of the Square Hoot of Polynomials. 

1st. A binomial can never be a perfect square. For, its root 
cannot be a monomial, since the square of a monomial will 
be a monomial ; nor can its root be a polynomial, since the 
square of the simplest polynomial, viz., a binomial, will cou 
tain at least three terms. Thus, an expression of the form 

a2±62 

can never be a perfect square. 

2d. A trinomial, however, may be a perfect square. If so, 
when arranged, its two extreme terms must be squares, and the 
middle term double the product of the square roots of the other 
two. Therefore, to obtain the square root of a trinomial, when 
it is a perfect square. 

Extract the square roots of the two extreme terms, and give these 
roots the same or contrary signs, according as the middle term is 
vositive or negative. To verify it, see if the double 2^roduct of thi 
two roots is equal to the middle term of the trinomial. 

Thus, 9a^ — 48a*62 -f 64a26* is a perfect square, 
for, y^9^ =z 3a3 ; and, ^^64^2^4 = _ 8^52 . 
also, 2 X 3a3 x{— SaP) = — ^SaW^, the middle term. 

But 4a2 + Uab + 962 

is not a perfect square : for, although 4^2 and + 9^2 are per- 
fect squares, having for roots 2a and 36, yet 2 x 2a x 36 m 
not equal to 14a6. 

Of Radical Quantities of the Second Degree. 

102* A radical quantity is the indicated root of an imperfect 
power of the degree indicated. Radical quantities aiie some- 
times called irrational quantities, sometimes surds, but mora 
commonly, simply radicals. 

The indicated root of a perfect power of the degree indi 
cated, is a rational quantity expressed under a radical form. 



134: ELEMENTS OF ALGEBRA. [CHAP. V. 

An indicated square root of an imperfect square, is called 
tt radical of the second degree. 

An indicated cube, root of an imperfect cube, is called a radi- 
cal of the third degree. 

Generally, an indicated n^^ root of an imperfect n^^ power, 
is called a radical of the n*^ degree. 

Thus, .yf^i v'^ ^^^^ -v/^' are radicals of the second degree; 
IJ 4, ^/Ts" and ^/TI, are radicals of the third degree; 

and \/~^y \f^ ^^^^ '\/^5 ^^® radicals of the n^^ degree. 
The degree of a radical is denoted by the index of the 
root. 

The index of the root is also called the index of the radical. 

103» Since like signs in both factors give a plus sign in thw 
product, the square of — a, as well as that of + <^5 will be 
a^ : hence, the square root of a^ is either -{-a or — a. Also, 
the square root of ^ba^^ is either -f- bab"^ or — bah"^. Whence 
we may conclude, that if a monomial is positive, its square root 
may be affected either with the sign + or — ; 

thus, V^^ ~ ^ ^^^' 

for, + Sa^ or — Sa^, squared, gives 9a^. The double sign rt 
with which the root is affected, is read plus or minus. 

If the proposed monomial were negative^ it would have ne? 
square root, since it has just been shown that the square of every 
quantity, whether positive or negative, is essentially positive. 

Therefore, such expressions as, 

are algebraic symbols which indicate operations that cannot be 
performed. They are called imaginary quantities^ or rather, 
imaginary expressions, and are frequently met mth in the so- 
lution of equations of the second degree. Generally, 

Every indicated even root of a negative quantity is an imaginary 
expression. 

An odd root of a negative quantity may often be extracted 
Fo* example, ^"^^27 = - 3, since (- 3)^ == — 27. 



CHAP, v.] RADICALS OF THE SECOND DEGREE. 136 

Radicals are similar when they are of the same degree and 
the quantity under the radical sign is the same in both. 

Thus, GiJ~b and cV^ are similar radicals of the second 
degree. 

Of the Simplijicaiion of Radicals of the Second Degree, 

104. Radicals of the second degree may often be simplified, 
and othermse transformed, by the aid of the following prin- 
ciples. 

1st. Let the .^/a^ and ,V^ denote any two radicals of the 
second degree, and denote their product by p\ whence, 
/^x/b=p . .. . . (1). 

Squaring both members of equation (1), (axiom 5), we have, 

or, ab^'p^ - - - - (2). 

Extracting the square root of both members of equation (2), 
(axiom 6), we have, 

but things which are equal to the same thing are equal to eacli 
other, whence, 

J~Gi y. J^ — J~G^ \ hence, 

The product of the square roots of two quantities is equal to 
the square root of the product of those quantities, 

2d. Denote the quotient of .^fa by y^ by q ; whence, 



yr - ■ • • ■ «• 

Squaring both members of equation (1), we find, 






= 9\ 



or, . ^ = q^ .... (2). 

Extracting the square root of both members of equation (2), 
we have, 

a 

j = q. 



136 ELEMENTS OF ALGEBKA. [CHAP. V. 

Things which are equal to the same tiling a e equal to each 
otlier, whence, 



^=\lt' hence, 

The quotient of the square roots of two quantities is equal to 
tJie square root of the quotient of the same quantities. 

105. The square root of 98a6* may be placed under the foria 
y 98a6^ == ^4Qb^ X 2a, 
which, from the 1st principle above, may be ^mtten. 



y496^ X ./2a = 7b^/2a. 
In lil^e manner. 



y/ 4:5a%h^d —^^aWc^ X 5>bd = Sahc^ obd. 
y/864^^255^ =y i44a2^/^cio X 6bc = VZab\^/'^. 
'The quantity which stands without the radical sign is called 
tlie co-efficient of the radical. 

Thus, 752^ 3a5c, and 12ab'^c^, are co-efficients of the radicals. 
In general, to simplify a radical of the second degree : 
I. Resolve the quantity under the radical sign into two factors^ one. 
of which shall be the greatest perfect square which enters it as a factor. 

II. Write the square root of the perfect square before the radical 
sign., under vjhich place the other factor. 

EXAMPLES. 



1. Eeduce /ibcfibc to its simplest form. 

2. Reduce Vl28Pa^ to its simplest form. 

3. Reduce / Z2a%'^c to its simplest form. 

4. Reduce J ^b'oa^b^c^ to its simplest form. 

5. Reduce J \024.a%\^ to its simplest form. 

6. Reduce V728a^Pc^ to its simplest form. 

If the quantity under the radical sign is a poljnomiai, we 
may often simplify the expression by the same rule. 



CHAP, v.] RADICALS OF THE SECOND DEGREE. 137 

Take, for example, the expression 

^' ci?h + ^d^h^ 4- 4a63. 
The quantity under the radical sign is not a perfect square : 
but it can be put under the form 

ah (a2 -I- 4a6 + W). 
Now, the factor within the parenthesis is evidently the square 
of a -f 26, whence we have 

ya36 + 4a262 4.4ap = (a + 26) ^/^^ 
105*» Conversely, we may introduce a factor under the radical 

sign- 
Thus, aJ^^^J~a^J^^ 

which by article 104, is equal to J v?h Hence, 

The co-efficient of a radical may he passed under the radical sign, 
as a factor, hy squaring it. 

The principal use of this transformation, is to find an ap- 
proximate value of any radical, which shall differ from its true 
value, by less than 1. 

For example, take the expression 6^13. 

Now, as 13 is not a perfect square, we can only find an ap- 
pix)ximate value for its square root ; and when this approximate 
value is multiplied by 6, the product will differ materially from 
the true value of 6y^l3. But if we write, 

6^13 r:ry/62 X 13 = ^ 36 X 13 =^468, 
we find that the square root of 468 is the whole number 21, 
to within less than 1. Hence, 

6VT3 = 21, to within less than 1. 
In a similar manner we may find, 

12 /T =31, to within less than 1. 

Addition and Subtraction. 

106i In order to add or substract similar radicals : 

Add or suhtract their co-efficients, and to the sum or differ^ 
€7Ke annex the common radical. 



138 ELEMENTS OF ALGEBRA. [CHAP. V 

Thus, 3a^ + bc^T = (3a + 5c)^; 

and 3a^ -^c^'^ = (3a - Sc)^^ 

In like manner, 

l^a + 3 /2^ = (T + 3)^2^ = 10^ ; 

and "^V^ ~ "^ -/^ = — 3)^2^ = 4^2^. 

Two radicals, which do not appear to be similar, may becomi 
so bj simplification (Art. 104). 
Por example, 

y/4S^+ b,/lb^ =.: 46^3^+ 55y3a = 9^*^37; 

Alsa; 2 ^45 - 3y~5~ = 6 ^5" - 3 ^5" = 3 ^5. 

When the radicals are not similar, their addition or subtrac- 
tion can only be indicated. 

Thus, to add 3w^ to SV^, we write, 

577+3/6. 

Multiplication of Radical Quantities of the Second Degree. 

107. Let a.J~b and c/~d denote any two radicals of the second 
degree ; their product will be denoted thus, 

which, since the order of the factors may be changed without 
altering the value of the product, may be wi-itten, 

axe Xy^ x/Z 
The product of the last factors from the 1st principle of Art. 
104, is equal to ^ hd\ we have, therefore, 

a^ -X c^= ac^bd. 

Hence, to multiply one radical of the second degree by au 
ether, we have the following 

RULE. 

Multijdy the co-efficients together for a new co-efficient; after this 
write the radical sign, and imder it the product of the quantities 
under both radical signs. 



CHAP, v.] RAEICALS OF THE SECOND DEGREE. 139 



EXAMPLES. 



2. 2ay^ X Sa^c = U^,J1K^ = QaHc. 



3. 2a^ o? + b"^ X - 3ayaM^52 ^ _ q^2 (^2 ^ y.y 

Division of Radical Quantities of the Second Degree. 

108* Let OL-Jh and c.Jd represent any two radicals of the 
second degree, and let it be required to find the quotient of the 
first by the second. This quotient may be indicated thus, 

-^^-— r, which is equal to — X "^ • 



but from the 2d principle of Art. 104, 

'/^ _ /T «y^ a fh 

/' — ^ ~\/ Ti hence — P= = — \/ —• 

d yd' c^ c V d 

Hence, to divide one radical of the second degree by another, 
we have the following 

' RULE. 

Divide the co-efficient of the dividend by the co-efficient of the 
divisor for a new co- efficient ; after this, write the radical sign^ 
placing under it the quotient obtained by dividing the quantity under 
the radical sign in the dividend by that in the divisor. 

For example, ^ajb • 2hJ~c — -^.sj — ; 
' ^ /to V c 

And, \2ac^^bc -f- ^c^b — 3(z \J -^ — 3a,^/3e.~ 



26 

109. The following transformation is of frequent application ia 
finding an approximate value for a radical expression of a par- 
ticular form. 

Having gifen an expression of the form, 

a a 

or 



V + v"^ P — yfi 



/ 

140 ELEMENTS OF ALGEBRA. [CHAP. V 

in which tt and p are any numbers whatever, and g not a per 
feet square, it is the object of the transformation to render the 
denominator a rational quantity. 

This object is attained by multiplying both terms of the frac- 
tion by j5 — -/^ when the denominator is jt? -f v^' ^^^ ^7 
V-\-J~qi when the denominator is p—^q\ and recollecting 
that the sum of two quantities, multiplied by their difference, is 
equal to the difference of their squares : hence, 

P + -y/l (p + V^)(i^-v/?) P^-^ P^-Q 

a _ CLJP + -y/g) _ ajp + V~q) _ gj? + «V^ 

P-yf~9. {P - s/a) {P -^ -sT^) P'^-9. f'-q 

in which the denominators are rational. 

As an example to illustrate the utility of this method of ap- 
proximation, let it be required to find the approximate value of 

7 

the expression — . We write 

3 — ^5 

7 _ 7(3 + ^f^ _ 21 + 7 y^ 

3 - ^~h " 9 - 5 ~ 4 

But, ^^= ^49 X 5 = ,/245 = 15 to within less 

than 1. Therefore, 

7 21 + 15 to wdthin less than 1 



3-/5" 



= 9 to within 



less than — ; hence, 9 differs from the true value by less than 
one four til. -^ 

If we wish a more exact value for this expression, extract the 
square root of 245 to a certain number of decimal places^ add 21 
to this root, ana divide the result by 4. 

7 ^/5^ 
Take the expression, — — — :, 

aud find its value to within less than 0.01. 



nHAP. v.] EXAMPLES IN THE CALCULUS OF RADICALS. 141 

We have, 

7-/5 7v^(-v/Tl - -/ 3y_ 7/55 - TylS 

/rr-i-/3~ 11-3 ~ 8 

Now, 7/55 =/55 X 49 =:/2695 = 51.91, within less than 0.01, 
and 7yr5=/l5x49:=:y/735 z= 27.11; - - . ; 

therefore, 

7a/5~ 51.91-27.11 24.80 „,^ 

/n+/3 8 8 

Hence, we have 3.10 for the required result. This is true to 

within less than — -:. 

By a similar process, it may be found, ihat, 

o _l_ o -. /t" 

— — —=2.123, is exact to within less than 0.001, 

5/12-6/5 

Remark. — The value of expressions similar to those above, 
may be calculated by approximating to the value of each of the 
radicals which enter the numerator and denominator. But as 
the value of the denominator would not be exact, we could 
not determine the degree of approximation which would be 
obtained, whereas by the method just indicated, the denomina- 
tor becomes vrational^ and we always know to what degree of 
accuracy the approximation is made. 

PROMISCUOUS EXAMPLES. 



1. Simplify ya25: Ans. 5/5. 

/5Q 

2. Reduce v/rrz; ^^ its simplest form. 

We observe that 25 will divide the numerator, and hence, 



147 
|e Since the perfect square 49 w'll divide 14*7, 

I 



142 ' ELEMENTS OF ALGEBEA. [CHAP. V. 

Divide the coefficient of the radical by 3, and mii tiply the num 
ber under the radical by the square of 3 ; then, 



5 /2 5/18 5 /— 



3. Reduce -/ 98a'^x to its most simple form. 

Ans. 7a^ 2x, 

4. Reduce ^ (x^ — a^x'^) to its most simple form. ■ ^ 

5. Required the sum of ^72 and Vl28. 

Ans. 14^ 

6. Required the sum of .V27 and ^ 147. 

A71S. lO^/W. 



/2~ 

7. Required the sum of \/ — and 

""''• 30 

8. Required the sum of 2^025 and 3.^646^ 

9. Required the sum of 9^243 and 10^363. 

/3~ /~^ 

10. Required the difference of \/— aad \/^« 

Ang. 1/15 

11. Required the product of 5.V8~ and 3V5. 

Ajis. 30.^/Ia 

2 /T~ 3 /t" 

12. Required the product of -W \/ ~o' ^^^^ TvTo* * 

Ans. 1/35. 

13. Divide 6/TO by 3y^ 

14. What is the sum of ^48^ and h^^a. 

15. What is the sum of /iSa^^a and /SOa^fis. 

^7?5. (3a26 + 5a6)y^X 



CHAPTER VI. 



EQUATIONS OF THE SECOND E EGllEE. 



110» Equations of the second degree may involve but 0710 
unknown quantity, or they may involve more than one. 
We shall first consider the former class. 

Ill, An equation containing but one unknown quantity is 
said to be of the second degree^ when the highest power of 
the unknown quantity in any term, is the second. 

Let us assume the equation, 

__ a;2 — ex -\- d =: cx^ -\ — -x -{• a. 
a 

Gearing of fractions, 

adx"^ — bcdx + hd"^ =z bcdx^ + b^x + (^bd 

transposing, adx^ — bcdx^ — bcdx — b^x =: abd — bd"^ 

factoring, (ad — bcd)x^ — [bed -f- b'^)x = abd — bd^ 

dividing both members by the co-efficient of x^, 

, bed + 52 abd - bd^ 



.X z= 



ad — bed ad — bed ' 

If we now replace the co-efficient of x by 2p, and th« 
second member by q, we shall have 

x"^ -f 2j)x =: q ; 
and since every equation of the second degree may be reduced, 
in like manner, we conclude that, every equation of the second 
degree, involving but one unknown quantity, can be reduced to 
tlie form 

ar2 -f 2px — gr, 
by the following 



144 ELEMENTS OF ALGEBRA. [CHAP. VI. 

RULE. 

I. Clear the equation of fractions ; 

II. Transpose all the known terms to the second member^ and 
all the unknown terms to the first. 

III. Reduce the terms involving the square of the unknown 
quantity to a single term of tiuo factors, one of which is the 
square of the unknown quantity ; 

IV. Then, divide both members by the co-efficient of the square 
of the unknown quantity. 

112. If 22?, the algebraic sum of the co-efficients of the first 
powers of x, becomes equal to 0, the equation will take the 

form 

x^ z= q, 

and this is called, an incomplete equation of the second degree. 
Hence, 

An incomplete equation of the second degree involves only the 
second power of the unknown quantity and known terms, and ma^ 
be reduced to the form 

x^ = q. 

Solution of Incomplete Equations. 

113. Having reduced the equation to the required ' form, we 
have simply to extract the square root of both members to find the 
value of the unknoivn quantity. 

Extracting the square root of both members of the equation 

x"^ z=z q, we have x = -/^ 

If 5' is a perfect square, the exact value of x can be found 
by extracting the square root of q, and the value of x will then 
be expressed either algebraically or in numbers. 

If q is an algebraic quantity, and not a perfect square, it m.ust 
be reduced to its simplest form by the rules for reducing radi- 
cals of the second degree. If 5- is a number, and not a perfect 
square, its square root must be determined, approximately, by 
the rules already given. 



CHAP. VI. J EQUATIONS OF THE SECOND DEGREE. 146 

But the square of any number is -f-, whether the number 
itself have the -{-or — sign; hence, it follows that 

{-\-^Y = g., and i-^Y = q] 
and therefore, the unknown quantity x is susceptible of two dis- 
tinct values, viz : 

and either of these values, being substituted for x, will satisfy 
the given equation. For, 

and ^^ = — -y/? X — -1/^= ^5 hence. 

Every incomplete equation of the second degree has two roots 
which are numerically equal to each other ; one having the sign 
plus, and the other the sign minus (Art. 77). 



EXAMPLES. 

1. Let us take the equation 

which, by making the terms entire, becomes 

Sx^ - 72 + 10^2 ^ 7 _ 24a;2 + 299, 
and by transposing and reducing 

42;r2 =: 378 and x^ = -—- = 9 ; 

42 

hence, x = -|- V9^=: + 3 ; and x = — V9*= — 3, 

2. As a second example, l*^t us take the equation 

Sx^ =: 5. 
Dividing both members by 3 and extracting the square root, 

In which the values of x must be determined approxinr. ai.«ly 

3. What are the values of x in the equation 

11(.^2 - 4) 1= 5(a;2 + 2). Ans. a: = zb 3. 

4. What are the values of x in the equation 

-i/m2 — x^ m 
= n. Ans X = ±: 



yr+ 



146 ELEMENTS OF ALGEBRA. [CHAP. VI 

Solution of Equations of the Second Degree. 

114» Let us now solve the equation of the second degree 

a;2 + ^px zzzq. 
If we compare the first member with the square of 
x-^-jp^ which is a;^ -f-2pa: -j-^^^ 
we see, that it needs but the square of "p to render it a perfect 
square. If then, jp2 \y^ added to the first member, it will be 
come a perfect square ; but in order to preserve the equality of 
the raiembers, jp^ must also be added to the second member. 
Makiiig these additions, we have 

x^ + 2pa; ■\- jf^ z= q -\- p^ 'y 
this is called, completing the square, and is done, hy adding tke 
square of half the co-eficient of x to hath members of the equa 
Hon, 

Now, if we extract the square root of both members, we have, 



x-\-p= :^^q +p% 
and by transposing p, we shall have ' 

X = —p -h^q 4-Jo2, and x = —p —^q-]-p\ 
Either of these values, being substituted for x in the equation 

x^ + 2px = q 
will satisfy it. For, substituting the first value, 

x^ = ( —p +^q-{-p^f =p^ — 2p^q+p^ -hq-i p\ 
and 



2px = 2px{-p +^q-\-p^) =:-2p^ + 2p^q+p^, 

by adding x^ -\- 2px = q. 

Substituting the second value of x, we find, 

a;2 = ( —p —^qJ^f.f — ^2 _|. 2p^ q -\- p"^ -\- q -f ;?% 
and 



'ilpx==2p{-p-./q^^) =-2p^-2p^q+p''; 

by adding x^ + 2px = q ; 

and consequently, both values found above, are roots of the 
equation. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGKEB. 147 

In order to refer readily, to either of these values, we shall 
call the one which arises from using the -j- sign before the 
radical, the first value of a;, or the first root of the equation; 
and the other, the second value of x^ or the second root of the 
equation. 

Having reduced a complete equation of the second degree to 
the form 

x^ + ^px = q, 
we can write immediately the two values of the unknown quan 
tity by the following 

RULE. 

I. The first value of the unknown quantity is equal to half 
the co-efficient of a:, taken with a contrary sign^ plus the square 
root of the second member increased by the square of half this 
co-efficient. 

II. The second value is equal to half the co-efficient of x, 
taken with a contrary sign, minus the square root of the second 
member increased by the square of ' half this co-efficient. 





EXAMPLES. 


1. Let us 


take as an example, 




x'^ — lx-\- 10 = 0. 


Reducing 


to required form. 




a;2 _ 7^ ^ _ 10 ; 



7 / 49 

whence by the rule, a; = — + \/— 10 4-^ = 5; 

and, « = |--\/-10 + ^ = 2. 

2. As a second example, let us take the equation 

¥"--2" + T = ^-¥^-^ +12- 



148 



ELEMENTS OF ALGEBRA. 



[CHAP. VI 



Reducing tt the required form, we have, 



.2. A -^ 
' "^22^^" 22 



whence, «^ = " ^ +\/^ + (4)' 



and a; = 

22 

It often occurs, in the solution of equations, that p^ and q 
are fra^ctions, as in the above example. These fractions most 
generally arise from dividing by the co-efficient of x^ in the 
reduction of the equation to the required form. When this is 
the case, we readily discover the quantity by which it is neces- 
sary to multiply the term q, in order to reduce it to the 
same denominator with p^ ; after which, the numerators may be 
added together and placed over the common denominator. 
After this operation, the denominator will be a perfect square, 
tmd may be brought from under the radical sign, and will 
become a divisor of the square root of the numerator. 

To apply these principles in reducing the radical part of the 
vdues of X, in the last example, we have 



/mo . / 1 \2 ^ /360 X 
V-22-^y=^V-(22)^ 



X 22 



+ 



= *4v/ 



•921 = 



(22f 
89 



/79i 

V-7 



7920 + 1 



(22)= 



wid therefore, the two values of x become, 
^ ~" 22 "^ 22 ~ 22 ~" '* 



and 



*~ ~22~'2" 



90 
22 



45 
11 



either of which being substituted for x in the given equation, 
will satisfy it. 

3. What ar« the valuts of a; in the equation 
ax^ — ac := ex — bx^ 



CHA.P. VT.] EQUATIONS OP THE SECOND DEGREE. I4fl 

Eeducing to required form, we have, 

c ac 



a-\-b a-fJ ' 



C I CLC (j^ 

whence, x = -\- -— — — — ■ + \ / — — - + —- — — — , 

c / oc c 

and, a; = + -— ; — -^ — \/ — -7- -f 



2{a-\-b) W a + b 4 (a + bf 
Reducing the terms under the radical sign to a coiiiinoo 

denominator, we find, 

riTc ^ c2 _ Ma'^c 4- '. '^-^ -I- c2 _ y'4a2c + 4a6c + o^ ^ 

VM^'^4(a + i)2"~V"~4vc6+~6)2 -; 2 (a + 6) ' 

. c ± -i/ 4a2c 4- 4aZ>c -f- c^ 
hence, ^ = ^1^^^^^) • 

4. What are the values of x, in the equation, 

6a;2 — 37ar = - 57. 
By reducing to the required form, we have, 
2_37 __57 

"^ 6 "^ ~ 6 ' 



, 37 / 57 . /37\2 



Reducing the quantities under the radical sign to a com moo 
denominator, we have, 



^12 V (12)2 ^ (12)2* 
But, 114 X 12 = 1368 ; and (37)2 _ 13^9 . 



, , 37 , /-1368 + 1369 37 

hence, x =z + — r d= \ / ; — r = A 

^12 V a2 2 ^ 12 



(12)2 ~ ' 12 12' 



, 37 1 19 

or, X = -\ = — 

' ^ 12 ^ 12 6 ' 

37 1 

'^'»' " = + T2-T2 = ^- 

5. What are the values of x, in the equation, 
4a2 — ^x"' + 2ax = 18a5 — 186«. 



150 ELEMENTS OF ALGEBEA. (CHAP. VI 

Reducing to the required form, we have, 
x^ — ax = 2^2 _ 9ab -f 9b^ ; 



whence, x = -^ dt yj 2a^ — 9ab -{- db"^ +^ 



-|--\/¥-- + 



962. 



The radical part is equal to — 35 ; hence, 

Find the values of a; in the following 

EXAMPLES. 



a^ a , b 2x^ , a 

— -—x=l X -. Ans, x= --, 

6 b a 3 


_ b 
~ a* 


dfa; 8a;2 1 + c x^ ^ X 
c ^ 4 + ^ = c 4 + ^* 




Ans. a;=l 


_ d 
~ c 



4 T"^ 8 ~ 4 3* 



^m. a: = -, ^ = -J 



A 


90 90 27 


^?is. ar =i'. 4, a; = — — 




a; a; 4- 1 ~ « + 2' 


5. 


2a; -10 ^ a: + 3 

8 _ a; ■ '^ a; — 2* 


4 
^ns. a; = 7, a; = ~ 


6. 


fl^ _ 4. 5 - — T— ^^ 4- — a?. 
6» b a 


6 
^W5. a; =s'*, a; = 


7. 


a — b . 3a;2 a2 j 4. ^j 
e "+ 2 c2- c ^ 


a;2 62 * 

■^ 2 c2* 




-4n5. 


6 + a 6 — a 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE 
Ans. X = 



151 



"Y/ mri 



mn 



X = 



yf^-yf^ yf^^yf^^ 



, , 6a2 ^2^; ah - 2^2 3^2 
9 ahx^ r- 4 = 5— a:. 



2a — 6 3a + 26 

^/is. a; == , X ^=1 :- . 

ac be 



.^ 4a;2 2aj , ,^ ^^^ 3.t;2 , bSx 
10 -^ + — + 10 = 19 - -- + -^. 



a; 4- a q — x 

11. jLZJ:_5^" 



^?is. a; = 9, a; r= — 1. 



Ans. 



6-2' 



11 



(z 4- ^ 
12. 2a; -f 2 = 24 — 5a; — 2a;2. ^^s. a; = 2, 

Ans. a; = 15, and x^ — 14. 
Ans. a; = 5, and x — — 5|. 



13. a;2 _ a; - 40 = 170. 

14. 3a;2 -f 2a; - 9 = 76. 




V 



15. a2 + 62 - 25a; + .i-2 - 



Ans. X = 






Problems giving rise to Equations of the Second Degree mwlv- 
ing hut one unknoiun quantity. 

1. Find a number such that three times the number added t» 
twice its square will be equal to 65. 

Let x denote the number. Then from the conditions, 

2a;2 + 3a;=65 - - - (1) 



Whence, 
rerjucing 



65 _9_ 



3 /6 



a; =^ 5 and 



13 
1^' 



152 ELEMENTS OF AT GEBRA. ICHAP IV, 

Botli of these roots verify the equation : for, 

2 X (5)2 -f 3 X 5 = 2 X 25 + 15 == 05 ; 

A o / i3\2 "13 169 39 130 ^„ 

and 2(-_)+3x-- = — -- = — = 6o. 

The first root satisfies the conditions of the problem as enim 
elated. 

The second root will also satisfy the conditions, if we regard 

lis algebraic sign. Had we denoted the unknown quantity by 

— X, we shouleK'Iiave found 

2a;2-3a; = 65 - - - (2) 

13 
from which x =z — and x = ~ 5. 

We see that the roots of this equation differ from those of 

e<][uation (1) ^ly in their signs, a result which was to have 

been expectja, since we can change equation (1) into equation 
(2) by simply changing the sign of a;, and the reverse. 

2. A person purchased a number of yards of cloth for 240 
cents. If he had received three yards less, for the same sum, it 
would have cost him 4_ cents more per yard. How many yards 
dic^j^ purchase 1 

denote the numb^i^p^^^^BiDurchased. 




numb^^j^^^^^Hpi 
Lenote the nu^^r o 



ill denote the nunmer or cents paid per yard. 

received three yards less, 
X — 3, *\YOuld have denoted the number of yards purchased, and 
240 ,. , . . . ., . . . . .. 



,_3' 


IXC* V 1^ VJ.Oi.lV/Lt/\J. LiJ 


.t/ XI u.i.ii. u t^i yji. ^.i^j-iuo xx^ ijtiivA upj y a*\tu 


From the conditions of the problem, 




240 
x-S 


240 ^ 


by reducing. 


x^- 


3.-^ = 180, 


whence. 


x = l^^ 


^d a; = — 12. 


The value 


s^^H 


^■he conditions (f the problem, 


understood in 


their arit^^B 


Wl sense ; for. 11 yards for 240 



^^ 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 153 

240 

cents, gives — — , or 16 cents for the price of one yard, and 

ID 

12 yards for 240 cents, gives 20 cents for the price of one 
yard, which exceeds 16 by 4. 

The value -\- x = — 12, or — a; = -f 12, will satisfy the 
couditions of the following problem : 

A person sold a number of yards of cloth for 240 ^.enfs : 
if he had received the same sum for 3 yards more, it would 
have brought him 4 cents less per yard. How many yards did 
he sell? 

If we denote the number of yards sold by x, the statement of this 
last problem, and the given one, both give rise to the same equation, 

x^ —Zx = 180, 

hence, the solution of this equation ought to give the answers 
to both problems, as we see that it does. 

Generally, when the solution of the equation of a problem 
gives two roots, if the problem does not admit of two solu- 
tions there is always another problem whose statement gives 
rise to the same equation as the given one, and in this case 
the two roots form ans\^|||to both problems. 

3. A man bought a hor^^Rhich he sold for 24 dollars. At 
the sale, he lost as much per cent, on the price of his pur- 
chase, as the horse cost him. What did he pay for the horse? 

Let X denote the number of dollars that he paid for the horse : 

then, X — 24 will denote the number of dollars that he lost. 

But as he lost x per cent, by the sale, he must have lost 

X 

r:-—- upon each dollar, and upon x dollars he, lost a number 

of dollars denoted by — — ; we have then the equation 

a;2 
— -^.T— .24, whence a:^ — lOO.r = — 2400 ; 

Therefore, x =z 60 and x = 40. 

Both of these values satisfy the conditions of the problem. 



154 ELEMENTS OF ALGEBRA. [ClIAP. VL 

For, in the first place, suppose the man gave 60 dollars for 
the horse and sold him for 24, he then loses 36 dollars. But, 
from the enmiciation, he should lose 60 ^er cent, of 60, that is, 

Too -^^^^ =-100- = ^^' 

therefore, 60 satisfies the problem. 

K he pays 40 dollars for the horse, he loses 16 by the sale ; 

for, he should lose 40 per cent, of 40, or 

40 
40X^^ = 16; 

therefore, 40 satisfies the conditions of the problem. 

4. A grazier bought as many sheep as cost him £60, and 
after reserving 15 out of the number, he sold the remainder 
for £54, and gained 2^. a head on those he sold : how many 
did he buy? Ans. 75. 

5. A merchant bought cloth for which he paid £33 15s., which 
he sold agam at £2 85. per piece, and gained by the bargain 
as much as one piece cost him : how many pieces did he buy ? 

Ans. 15. 



6. What number is that, whichjfc^ig divided by the product 
of its digits, the quotient ^vill ^^^Pl and if IS he added to 
it, the order of its digits will l^Je versed ? Ans. 24. 

7. Find a number such that if you subtract it from 10, and 
multiply the remainder by the number itself, the product will 
be 21. Ans. 7 or 3. 

8. Two persons, A and B, departed from different places at 
the same time, and traveled towards each other. On meeting, 
it appeared that A had traveled 18 miles more than B ; and 
that A could have performed B"s journey .m 15J days, but B 
would have been 28 days in performing A's journey. How 
far did each travel 1 j A 72 miles. 

' ( B 54 miles. 

9. A company at a tavern had £8 15s. to pay for their 
reckoning ; but before the bill was settled, two of them left 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 155 

the room, and then those who remained nad 10s. apiece more 
to pay than before : how many were there in the company 1 

Ans. 7. 

10. What two numbers are those whose difference is 15, and 
of which the cube of the lesser is equal to half their product! 

Ans. 3 apd 18. 

11. Two partners, A and B, gained $140 in trade : A's money 
was 3 months in trade, and his gain was $60 less than his 
"tock : B's money was $50 more than A'^ and was in trade 5 
months: what was A's stock? Ans. $100. 

12. Two persons, A and B, start from two different points, and 
travel toward each other. When they meet, it appears that 
A has traveled 30 miles more than B. It also appears that 
it will take A 4 days to travel the road that B had come, 
and B 9 days to travel the road that A had come. What was 
their distance apart when they set out? Ans. 150 miles. 

Discussion of Equations of the Second Degree involving but 
one unknown quantity/. 



115. It has been shown t^H every complete equation of the 
second degree can be reduce^^ the form (Art. 113) 

x^-i-2px = q . . - (1), 

in which p and q are numerical or algebraic, entire vr frac- 
tional, and their signs plus or minus. 

If we make the first member a perfect square, by completing 
the square (Art. 112*), we have 

x^ + 2px -f- i?^ = 3' + i>^ 
which may be put under the form 

{x -\-pY = q -{-p^. 

Now, whatever may be the value of 5' + 7>^ its square root 
may be represented by m, and the conation put under the form 

{x -\- pY = m^, and consequently? {x + 2^)2 — m"^ — 0. 



156 ELEMENTS OF ALGEBRA. '"'" [CHAP. VI. 

But. as the first member of the last equation is the diiFerenc« 
between two squares, it may be put under the form 

(x -\- p —m) {x -{- p -{- m) = - - - (2), 
in which the first member is the product of two factors, and the 
second 0. No ,7, we can make this product eq^al to 0, and 
consequently satisfy equation (2) only in two different ways . 
viz., by making 

X -i- p — m = 0, whence, x =z — p -{- m, 
or, by making 

X -\- p -A- m ^ 0, whence, x = — p — m. 
Now, either of these values being substituted for x in equa- 
tion (2), will satisfy that equation, and consequently, will satisfy 
equation (1), from which it was derived. Hence, we conclude, 

1st. That every equation of the second degree has two roots, and 
only two. 

2d. That the first member of every equation of the second, degree^ 
whose second member is 0, can be resolved into two binomial fac- 
tors of the first degree with respect to the unknown quantity, having 
the unknown quantity for a first term and the tioo roots, with iheit 
signs changed, for second terms. 

For example, the equation 

a;2 + Sx -"^8 = 
being solved, gives 

X z=4: and x = —7 ; 

either of which values will satisfy the equation. We also have 

{x - 4) {x + 7) 1= x^ + 3a; - 28 = 0. 

If the roots of an equation are known, we can readily form 
the binomial factors and deduce the equation. 

EXAMPLES. 

1. What are the factors, and what is the equation, of whicli 
the roots are 8 and — 9 1 

Ans, X — S and x -{- 9 are the bincmial factors, 
and x^ -{- X — 72 = is ne equat on. 



i 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 157 

2. What are the factors, and what is the equation, of which 
the roots are — 1 and + 1 ? 

X -\- 1 and X — 1 are the factors, 

and rr^ — 1 = is the equation. 

3. What are the factors, and what is the equation, whose 
roots are 

7 + -y/ - 1039 7 - V - 1039 ^ 

16 16 



are the factors, 

and Sx^ — 7a; + 34 =: is the equation. 

/ 116i If we designate the two roots, found in the preceding 
article, by x' and x'', we shall have, 

ic' = — ^ + m, 
re" = — J) — m', 



or substituting for m its value y/ g -{- p^, 



x' = -i>+y^+i>2, 

Adding these equations, member to member, we get 
x' -\-x" = —2p', 

and multiplying them, member by member, and reducing, 

we find 

X X = — q. 

Hence, after an equation has been reduced to the form of 

x^ -\- 2px z=z q 

1st. The ilgehraic sum of its two roots is equal to the co-effi- 
cient of the first power of the unknown quantity, with its sign 
changed. 

2d. The product of the two roots is equal to the aecond membor, 
with its sign chan^-ed. 



158 ELEMENTS OF ALGEBRA. [CHAP. VL 

If the sum of two quantities is given or known, their pro- 
duct will be the greatest possible when they are equal. 

Let 2/7 be the sum of two quantities, and denote their differ- 
ence by 2c?; then, 
p -\~ d will denote the greater, and p — d the less quantity. 

If we represent their product by q, we shall have. 
p^ — d^ =z q. 

Now, it is plain that q will increase as d diminishes, and 
that it will be the greatest possible, when c? = ; that is, when 
the two quantities are equal to each other, in which case the 
product becomes equal to p>^. Hence, 

3d. The greatest possible value of the product of ^ the two roots, 
is equal to the square of half the co-efficient of the first power 
of the unknown quantity. 

Of the Four Forms. 

117» Thus far, we have regarded p and q as algebraic quan- 
tities, without considering the essential sign of either, nor have 
we at all regarded their relative values. 

If we first suppose p and q to be both essentially positive, 
then to become negative in ^cession, and after that, both to 
become negative together, we^hall have all the combinations 
of signs which can arise. The complete equation of the second 
degree will, therefore, always be expressed under one of the 
four following forms : — 









x^ + Ipx = 


q 


(1), 








x^ - 2px = 


q 


(2), 








X^ + 2pX := 


-q 


(3), 








a;2 — 2px = 


-9. 


(4). 


These 


equations 

X 


being solved, 


give 






= -P^/~ 


q^p-" 


(1), 






X 


= +P±/~ 


q-Vp"- 


(2), 






X 


= -P^y/^ 


q+p"" 


(3), 






X 


= +p±^ 


q-{-p^ 


(4). 



CHAP. VI-] EQUATIONS OF THE SECOND DEGEEE. 159 

In the first and second forms, the quantity under the radical 
sign will be positive, whatever be the relative values of ^ and g-, 
since q and p^ are both positive ; and therefore, both roots 
will be real. And since 

q+p^yp"^, it follows that, ^ q+P^>P^ 
and consequently, the roots in both these forms will have the same 
signs as the radicals. 

In the first form, the first root will be positive and the 
second negative, the negative root being numerically the greater. 
_In the second form, the first root is positive and the second 
negative, the positive root being numerically the greater 

In the third and fourth forms, if 

p'^yq, 
the roots will -^^^ real, and since 



they will have the same sign as the entire part of the root 
Hence, both roots will be negative in the third fi)rm, arid both 
positive in the fourth. 

If p"^ — q^ , the quantity under the radical sign becomes 0, 
and the two values of a; m both the third and fourth forms 
will be equal to each other ; both equal to — ^j in the third, 
form, and both equal to -\- p m the fourth. 

If ^2 <^ q^ the quantity under the radical sign is negative, 
and all the roots in the third and fourth forms are imaginary. 

But from the third principle demonstrated in Art. 116, the 
greatest value of the product of the two roots is p"^^ and from 
'% the second principle in the same article, this product is equal 
to q ; hence, the supposition of p"^ <iq is absurd, and the values 
of the roots corresponding to the supposition ought to be im^ 
possible or imaginary. 

When any particular supposition gives rise to imaginary re- 
sults, we interpret these results as indicating that the suppo 
sition is absurd or impossible. 



160 ELEMENTS OF ALGEBRA. [CHAP. VL 

If p = 0, the roots In each form become equal with con- 
trary signs ; real in tho. first and second forms, and imaginary 
in the third and fourth. 

If q = 0, the first and third forms become the same, as also, 
the second and fourth. 

In the former case, the first root is equal to 0, and tne 
second root is, equal to — 2p ; in the latter case, the first root 
is equal to -\- 2p, and the second to 0. • 

If ^ = and q = 0, all the roots in the four forms reduce 
k) 0. 

In the preceding discussion we have made 
^2 > q^ p^ < q^ and f- — q\ 
we have also made p and q separately equal to 0, and then 
both equal to at the same time. 

These suppositions embrace every possible hypothesis that can 
be made upon p and q. 

118. The results deduced in article 117 might have been ob- 
tained by a discussion of the four forms themselves, instead of 
their roots, making use of the principles demonstrated in arti- 
cle 116. 

In the first form the product of* the two roots is equal to 
. — g-, hence the roots must have contrary signs ; their sum is 
— 2p, hence the negative root is numerically the greater. 

In the second form the product of the roots is equal to — q 
and their sum equal to + 2p ; hence, their signs are unlike, 
and the positive root is the greater. 

In the third form the product of the roots is equal to + <7 ; 
hence, their signs are alike, and their sum being equal to — 2/? 
they are both negative. 

In the fourth form the product of the roots is equal to + q^ 
and their sum is equal to -|- 2p ; hence, their signs are alike 
and both positive. 

If p z= 0, the sum of the roots must be equal to ; or the 
roots must be equal with contrary signs. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGKEE. 161 

If q = 0, the product of the roots is equal to ; hence, one 
of the roots must be 0, and the other will be equal to the co- 
efficient of the first power of the unknown quantity, taken with 
a contrary sign. 

If p = and q z= 0, the sum of the roots must be equaJ 
to 0, and their product must be equal to ; hence, the roots 
themselves n^st both be 0. 

119, There is a singular case, sometimes met with in the 
discussion of problems, giving rise to equations of the second 
degree, which needs explanation. 

To discuss it, take the equation 

ax^ -\- bx = c, 

,. , . —b ± V ^^ H- 4ac 
which gives x = — . 

If, now, we suppose a = 0, the expression for the value of 

X becomes 





-b:^b 

X — , whence. 



0' 



26 




But the supposition a = 0, reduces the given equation to 
bx z= c, which is an equation of the Jirst degree. 

The roots, found above, however, admit of interpretation. 

The first one reduces to the form — in consequence of the 

existence of a factor, in both numerator and denominator, which 
factor becomes for the particular supposition. To deduce the 
true value of the root, in this case, take 

_ — 6 + -v/ 62 + Aac 
^ ~ 2a ' 

and multiply both terms of the fraction by —b— ^b^4(icl 
after striking out the common factor — 2a we shall have 

_ 2c 

~6-fy62-h4ac' 
11 



162 ELEMENTS OF ALGEBRA. [CHAP. VT. 

ft 

in which, if we make a = 0, the -v^alue of x reduces to -=-^ 

the same value that we should obtain by solving the simple 
equation bx == c. 

The other root oo, is the value towards which the expression, 
for the second value of x, continually approaches as a is made 
smaller and smaller. It indicates that the equation, under the 
supposition, admits of but one root in finite terms. This should 
be the case, since the equation then becomes of the first degree. 

?^- 120t The discussion of the following problem presents most 
of the circumstances usually met with in problems giving rise 
to equations of the second degree. In the solution of this 
problem, we employ the following principle of optics, viz. : — 

The iiitenniy of a light at any given distance, is equal to its 
intensity at the distance 1, divided by the square of that distance. 

Problem of the Lights. 



C" A C B C 

121t Find upon the line which joins two lights, A and B, of 
different intensities, the point which is equally illuminated by 
tiie lights. ^ 

Let A be assumed as the origin of distances, and regard all 
distances measured from A to the right as positive. 

Let c represent the distance AB^ between the two lights ; 
a the intensity of the light A at the distance 1, and 6. the in- 
tensity of the light B at the distance 1. 

Denote the distance AC^ from A to the point of equal illu- 
mination, by x ; then will the distance from B to the same 
point be denoted by c — x. 

From the principle assumed in the last article, the intensity 
of the light u4, at the distance 1, being a, its intensity at the 

distances 2, 3, 4, &c., will be — , — , — , &c. ; hence, at the 

4 y 10 

distance x it will be expressed oy — r. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 163 

In liKe manner, the intensity of B at the distance c — x^ \% 

p r^ ; but, by the conditions of the problem, these two 

\c — X) 

intensities are equal to each other, and therefore we LAve the 

equation 

a h 



a;2 (c - XY ' 

which can be put under the form 
(c - xY b 



a 



c — ce ± -i/6~ - ^ 
nenee, = -~^ ; whcAice 



c -y/a 



('), 



(2). 



Since both of these values of x are alM'ays real, we conclude 
that there will be two points of equal illLunination on the line 
A B, or on the line produced. Indeed, it is plain that there 
should be, not only a point of equal illumination between the 
lights, but also one on the prolongation of the line joining the 
lights and on the side of the lesser one. 

To discuss these two values of x. 

First, suppose a ^ b. 

The first value of x is positive; and since 

VI <i 

V« + /* 

this value will be less than c, and consequently, the first point (?; 
will be situated between the points A and B. We see, moreover, 
that the point will be nearer B than A ', for, since a > o, we 
have 

y/a-\-^a or, 2^ a. >(y^4-y^)", whence 



— :=r^ — : > — ; and consequently, — z=:^ > — 



164 ELEMENTS OF ALGEBRA. LCHAP. VL 

The second value of x is also positive; but since 

-/" >i 

il will be greater than c; and consequently, the second pc 
wiL be at some point (7', on the prolongation of AB^ and 
the right of the two lights. 

This is as it should be; for, since the light at A is most 
intense, the point of equal illumination, between the lights, ought 
to be nearest the light B\ and also, the point on the prolonga- 
tion of AB ought to be on the side of the lesser light B. 

Second, suppose a <Cb. 
The first value of x is positive ; and since 

this value of x will be less than c; consequently, the first point 
will fall at some point C, to the right of A, and between A 
and B. 



C" A OB C 

We see, moreover, that it will be nearer A than B\ for, 
since a<Cb, we have 

-/a 4- V b > 2 V a, and consequently, ^ — — < — . 

The second value of x is essentially negative, since the nume- 
fator is positive, and the denominator essentially negative. 

We have agreed to consider distances from A to the right 
positive; hence, in accordance with the rule already established 
for interpreting negative results, the second point of equal illu- 
mination will be found at C'\ somewhere to the left of A. 

This is as it should be, since, under the supposition, the light 
at B is most intense; hence, the point of equal illumination, 
between the two lights, should be nearest A, and the point in 
the prolongation of AB, should be on the side nearest the 
feebler light A. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 166 

TJiird^ suppose a = 5, and <; > 0. 

Q 

The firs*, value of x is then positive, and equal to •— henc«^ 

the first point is midway between the two lights. 

The second value of x becomes ^ = oo , a result which in- 

dicates that there is no other point of illumination at a finito 
distance from A. 

This interpretation is evidently correct; for, under the supp{>- 
sition made, the lights are equally intense, and consequently, the 
point midway between them ought to be equally illuminated. 
It is also plain, that there can be no other point on the line 
which will enjoy that property. 

Fourth, suppose b ==.a and c = 0. 

The first value of x becomes, — = 0, hence the first point 

is at A. 

The second value of x becomes, — , a result which indicates 

that there are an infinite number of other points which arc 
equally illuminated. 

These conclusions are confirmed by a consideration of the con- 
ditions of the problem. Under this supposition, the lights are 
equal in intensity, and coincide with each other at the point A. 
That point ought then to be equally illuminated by the lights, 
as ought, also, every other point of the line on whidi the lights 
are placed. 

Fifth, suppose CL^ h, or a <^h, and c = 0. 

Under these suppositions, both values of x reduce to 0, whicli 
shows that both points of equal illumination coincide with the 
point A. 

This is evidently the case, for, since a is not equal to \ 
and the lights coincide at ^, it is plain that no other point than 
A can be equally illuminated by them. 

The preceding discussion presents a striking example of the 
precision with which the algebraic analysis responds to all tlie 
relations which exist between the quantities that enter a problem. 



106 ELEMENTS OF ALGEBRA. LCHAP. VL 

EXAMPLES INVOLVIXG RADICALS OF THE SECOND DEGREE. 

1. Given, x +-/a^ + x^ = , to find the values of x, 

^ y a2 + a;2 

By reducing to entire terms, we have, 

x^cC^ 4- ^^ + a^ + x^ = 2a2, 
oy transposing, ^-y/^^ + ^^ = a^ — rc^, 

and by squaring both members, a^x"^ -\- x* = a^ — 2a'^x'^ -{- x*, 
whence, Sa'^x'^ = a\ 



and, x=±:^^—.- 

2. Given, v/ -^ + ^^ — V ~i " ^^ = ^' *^ ^^-^ ^^® values of a:. 
By transposing, \/-:^ + ^^=\/-:5~"^^ + ^j 
squaring both members, — + i^ = — — ^^ + 25 W — — 6^ -j> J2 . 

whence, 6^ _ 26 w — — J^ and 6 z= 2 \/— — 6^ j 



4a.2 



squaring both members, 6^ = --^ 46^ ; 



4a2 / * 2a 



and hence, x^ = — , and a; = ± . 

gj /fl2 iK^ ^ 

8. Given, Y \/ — = -^, to find the values of x. 

X V a;2 h 

Ans. X = ± J 2ab — b^, 

4. Given,\/^^— -f 2\/ — ? — = b^\/ — ^ — , to tmd the 
v^ues of a;. .a 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 167 



n ^ / A»2 /p2 

5. Given, = b, to find the values of x. 

Ans, x= ±1 - — ,-j-. 
1 -i-b 



6 Given, — , to find the values oi x. 

^ X — ^ x — a X — a 

Ans. X = -^ -^. 

1 ± 2n 



7. Given, h — — — — =\/ ~? ^^ ^^^ t^® values of as. 



y^ V^ ^ ^ 



A71S. X — ±i 2y^ 



a6 - 62. 



8. Given, ■•> • ^ = b, to find the values of x. 

a-\- X 



^2b - 62 

Of trinomial Equations. 

122» A trinomial equation is one which involves only terms 
containing two difierent powers of the unknown quantity and a 
known term or terms. 

123. Every trinomial equation can be reduced to the form 

x'^ + 2px'^ — q (1), 

in which m and n are po^tive whole numbers, and p and q 
known quantities, by means of a rule entirely similar to that 
given in article 111. 

If we suppose m = 2 and ?i = 1, equation (1) becomes 
x^ + 2ipx =r g, ^ 

a trinomial equation of the second degree. 

124. The solution of trinomial equations of the second degree, 
has already been explained. The methods, there explained, are, 
with some slight modilications, applicable to all trinomidl equjk- 
tions in which m =. 2^, that is, to all equations of the form 

a;2n _|_ 2pic" = q. 



168 ELEMENTS OF ALGEBRA. [CHAP, VI. 

To demonstrate a rule for the solution of equations of this 
form, let us place 

a;" = y ; whence, x^"^ = y^. 
Those values of a;" and a:^", being substituted in the given 
equation, reduce it to ' 

y2 + 2py = g, 



whence, y = — p ±^q -hp"^, 

or, a;" = — ^ ± ^ ^ + p^. 

Now, the w'* root, of the first member, is x (Art. 18), and 
although we have not yet explained how to extract the n*^ 
root of an algebraic quantity, we may indicate the n*^ root of 
the second member. Hence, (axiom 6), 



x= y-i)±y7+. 



Hence, to solve a trmomial equation which can be reduced 
to the form a;^" -}- 2px" =: g, we have the following 

RULE. 

Reduce the equation to the form of x^^ -(- 2pa;" = g' ; the valuer 
of the unknown quantity will then he found by extracting the 
n*^ root of half the co-eficient of the lowest power of the un- 
known quantity with its sign changed^ ^9/^5 or minus the square 
root of the second member increased by the square of half the 
co-efficient of the lowest power of the unknown quantity. 

If n ^ 2, the roots of the equation are of the form 



= ^^J-p±^ 



q +ir 



"We see that the unknown quantity has four values, sirxe each 
of the signs + and — , which affect the first radical can be 
oombined, in succession, with each of the signs which affect the 
second ; but these values^ taken two and two, are numerically equals 
and have contrary signs. 



CHAP. VI.] TRINOMIAL EQUATIONS. 169 

EXAMPLES. 

1. Take the equation 

x^ — 2dx'^ = — IU. 
Tliis being of the required form, we have by appication of 
tlie rule, 



^=±\/|± 



/-^144 + «|^, 



/25 7 
whence, x= ±^^— dz — -, 

hence, the four roots are +4, — 4, +3, and — 3. 
2. As a second example, take the equation 



Whence, 


by 

X = 


the rule. 


X^ - 7rf2 =z 


:8. 






VI = 




-VI 


Q 




ty«+ T 


-4-' 



hence, the four roots are, 

+ 2/2, -2/2; +/^=n: and -./^^', 
the last two are imaghiary. 
3. X* — {2hc + 4a2) x'^=— hH'^. 



Ans. X = ±\/bc + 2^2 ± 2ay6c + a^^ 

4. 2a; - 7y^ = 99. Ans. x=Sl, x=^-^. 

5. 4- _ 6:r^ -h 4-:f23z 0. ^7... a; = ±. /f±^^^^5 

125* The solution of trinomial equations of the fourth degree 
requires the exfraction of the square root of expressions of the 
form of a db /b in which a and b we positive or negative, 

numerical or algebraic. The expression \/ a dtz Jh can some- 
times be reduced to the form of a' db ^fV or to the form 
J a" d= -/^" ; and when such transformation is possible, it is 



170 ELEMENTS OF ALGEBRA. [CHAP. TL 

advantageouu to effect it, since, in this case, we have only to 
extract two simple square roots ; whereas, the expression 



requires the extraction of the square root of the square root. 

To deduce formulas for making the required transformation, 
let us assume 



p-^q = ^a+/b .... (1), 

p-q = ^a-/b - (2); 

in which p and q are arbitrary quantities. 

It is now required to find such values for p and q as will 
satisfy equations (1) and (2). 

By squaring both members of equations (1) and (2), we have 
p2_^2pq-\-q^^a+/b. - . (3), 

p^ — 2pq -\-(f--=^a—Jh- - - (4). 
Adding equations (3) and (4), member to member, we get 

p^-^q^ = a (5). WUI^I 

Multiplying (1) and (2), member by member, we have,'*'' ' t 
^2 _ ^2 _ /a2 — J. 

Let us now represent J a^ — h by c. Substituting in the 
last equation, 

^2_^2^c .--*-. . (6). 
From (5) and (6) we readily deduce. 



p 



= ±y^ and , = ±y^., 



these values su])stituted for p and q^ in equations (1) and (2), 
give 






CHAP. VI.] TRINOMIAL EQUATIONS. 171 

I16IIC6 

ana ^^=.(^_^) . . (8). 

Now, if o? — h is a perfect square, its square root, c, will 
be a rational quantity, ai-id the application of one oi the for- 
mulas (7) or (8) will reduce the given expression to the re« 
quired form. If a^ — b is not a perfect square, the application 
of the formulas will not simplify the given expression, for, we 
shall still have to extract the square root of a square root. 

Therefore, in general, this transformation is not used, unless 
a^ — h is a perfect square. 



EXAMPLES. 



1. Reduce W 94 + 42^5 =v/ 94 -f- ^8820, to its simplest 
form. We have, a := 94, 5 =: 8820, 

whence, c = ^ a^ ~h = ^ 8836 -8820 = 4, 

a rational quantity ; formula (7) is therefore applicable to this 
case, and we have 

or, reducing, = ± {^/^ +-v/^^^ ' * 

hence, y^ + 42y^ = ± ( 7 -f 3^5). 

This may be verified; for, 

(7 -f 3y^)2 = 49 + 45 + 42^5 = 94 + 42y^. 

2. Reduce \/ np + 2wi^ — 2m,,Jnp + w^^, to its simplest 

form. We ha-'^e 

a — np -\- 2w2, and h = ^w}[np -{• m^), 
a^ - 5 m n2p2, and c — ^ a? — h = np\ 



^ 



172 ELEMENTS OF ALGEBRA. LCHAP. YL 

and therefore, formula (7) is applicable. It gives, 

-\V 2 V~ — 2 )• 

and^ reducing, ± (V^^i? + wi^ — ^tz). 

3. Reduce to its simplest form', 

4/16 + so^-i+y 16 - 3oy^. 

By applying the formulas, we find 



V 



16 + 30. 



1=5+3/ 



and 



y^l6-S0/^n 



1 = 5- 3/^^ : 
henas, -i/l6 + 30^"^=^ +\/l^ - ^^-/"^ = 



10. 



Tills example shows that the transformation is applicable to 
imaginary expressions. 

4. Reduce to its simplest form, 



y^28 + 10/3. 

5. Reduce to its simplest form, 



Ans. 5+^3. 



Ans. 2+y — 3. 



6. Reduce to its simplest form, 
y hc-\- '^h^bc-b'^ - \/bc — 26y^6c - 62. 

7. Reduce to its simplest form, 

k/ ab + 4c2 - cf2 — 2^4.abc^ - abd?. 

A?is. .y/~ob — V4c' — cP 



Ans. ± 2b 



CHAP, yi.] EQUATIONS OF THE SECOND DEGREE. 173 

Equations of the Second Degree involving two or more unknown 

quantities, 

126i Every equation of the second degree, containing two 
unknown quantities, is , of the general form 

ay2 + hxy + cx^ + dy ■\- fx + g =zO\ 
or a particular case of that form. For, this equation contains 
terms involving the squares of both unknown quantities, theij 
product, their first powers, and a 4vnown term. 

In order to discuss, generally, equations of the second degree 
involving two unknown quantities, let us take the two equations 
y£ the most general form 

ay"^ -{-hxy -\-cx'^ + dy ~\-fx-{-g =: 0, 
and a'y"^ -\- b'xy -f c'x"^ + d'y -\-f'x + ^' = 0. . 

Arranging them with reference to x, they become 
cx"^ + {hy -\-f)x-\-ay'^ + dy -^ g =0, 
c'x'' + {h'y +/) x + ahf + c?'y + ^' = ; 
from wliich we may eliminate 'ir^, after having made its co-efE- 
cient the same in both equations. 

By multiplying both members of the first equation by c', and 
both members of the second by c, they become, 

cc'x' ^{by^f)c'x-\-{ai/^-dy-^g )c' = 0, 
cc'x"^ + {b'y +/)c X -I- {a'y"^ + d'y + g')c ^ 0. 

Subtracting one from the other, member from member, we have 

[(be' — cb')y ■\-fc' — cf]x + {ac — ra')?/2 -f- (dc' — cd')y -f gc' 

- ^g' = 0, 

which give^ 

— (^^^ — cicyj'^ + (c d' — dc')y + cg^ — gc' 
^ - - {he' - cb')y -\-fc' - cf ^• 

This value being substituted for x in one of the proposed 
equations, will give a final equation^ involving only y. 

But without effecting the substitution, which would lead to a 
very complicated result, it is easy to perceive that the final 
equation involving y, will be of the fourth degree. For, the 



174 ELEMENTS OF ALGEBRA. [CHAF. VI: 

« 

numerator of the value of x being of the form 

my^ -\- ny -\r 'p, 
its square will be of the fourth degree, ani this square forms 
one of the parts in the result cf the substitution. 

Therefore, in general, the solution of two equations of the secona 
degree^ involving two unknown quantities, depends upon that of an 
equation of the fourth degree, involving one unknown quantity. 

127. Since we have not yet explained the manner of solving 
equations of the fourth degfee, it follows that we cannot, aa 
yet, solve the general case of two equations of the second 
degree involving two unknown quantities. There are, however, 
some particular cases that admit of solution, by the application 
of the rules already demonstrated. 

First. W^ can always solve two equations containing two 
unknown quantities, when one of the equations is of the second 
degree, and the other of the first. 

For, we can find the value of one of the unknown qua*: 
titles in terms of the other and known quantities, from the 
latter equation, and by substituting this in the former, we shall 
have a sii:|gle equation of the second degree containing but one 
unknown quantity, which can be solved. 

Thus, if we have the two equations 

a:2 + 2y2^22 - - - - (1), 
2x - y = \ . . . . (2), 
we can find from equation (2), 



whence, 



x" =. 



2 ' ' 4 

and by substituting this expression for x"^ in equation (1), we find 

1+^1! + 2,^ = 22; 

whence we get the values of y : that is, 

29 

2/ = 3 and y = _ — ; 

and by substitutmg in equation (2) we find, 
a? = 2 and x =z — —. 



oflAP. VI.] EQUATIONS OF THE SECOND DEGREE. 175 

t 

Second. We can always solve two equations of the second 
degree containing two unknown quantities when they are h&.M 
homogeneous wdth respect to these quantities. 

For, we can substitute for, one of the unknown quantiti^sa, 
an auxiliary unknown quantity multiplied into the second un- 
known quantity, and by combining the two resulting equations 
we can find an equation of the second degree, from which the 
value of the auxiliary unknown quantity may be determined, 
and thence the values of the required quantities can easily be 
found. 

Take, for example, the equations 

a;2 + rcy — 2/2 _ 5 . . . (j)^ 

Sx^ — 2xij—2ij^ = fy - - - (2). 

Substitute for y, px, jp being unknown, the given equat»~#ns 

become 

x^ -f- 'px^ — 'p^x^ = 5 - - - (3), 

3a;2 - 2p2;2 - 2^2^2 -^ e - - - (4). 

Finding the values of x^ in terms of ^, from equations (3) 

and (4), and placing them equal to each other, we deduce 

5 6 





Y-^-p—p^ ^ — 2p — 2p^ 


or reducing. 


p2-^4p = — ', 


whence, 


p = — , and i> = - y . 



Considering the positive value of p, we have, by substituting 
it i,n equation (3), 

or, x^ = 4', 

' whence, x = 2 and x = —2: 

and since y z=zpx we have y = 1 and y = — 1. 

Third There are certain other cases which admit of solution, 
but for wnich no fixed rule can be given. 

We shall illustrate the manner of treating these cases, ^j 
the solution of the following 



17^ ELEMENTS OF ALGEBRA. (CHAP. VL 



EXAMPLES. 



V- 



\. Given, -^ = 48/ 

X 

y )■ to find the values of x and y. 

^ = 24, 

Dividing the first by the second, member by member, we have 

=: 2, or J~y = 2 ; whence y = 4 ; 

' X 

y 

and by substituting in the second equation, we get 
.^=6, and a: = 36. 

2. Given, x J^f^J^y ^zi 19, ) 

'^ „ ,„„ r to nnd the values of x and y. 

Dividing the second by the first, member by member, we 
have 

But, x+^-\-y = \9: 

adding these, member to member, and dividing by 2, we find 

^ + y = i3, 

which substituted in the first equation, gives, 

/ xy — 6, or xy = 36, and x z=z — . 

■V y 

Substituting this expression for ^, in the preceding equation, 
we get, 







y 












or. 


y 


y' - i3y 


= — 


36; 


13 

~ 2 


± 




whence. 


=f*y- 


36 4- 


169 
4 


5 
2"* 


and finally. 




y = 9. 


or 


y = 


4; 






and since 




x-\-y = 


13, 










, 




x=4. 


or 


X = 


:9. 







CHAP. VI. J EQUATION'S OF THE SECOND DEGREE. 177 

3. Find the values of x and y, in the equations 
a;2 -f- 3:c + y = 73 — "Hxy 
y^ + Sy-{-x = M. 
By transposition, the first equation becomes, 
x^ 4- 2xy -{-Sx + y = 7S; 
to which, if the second be added, member to member, tHeie 
results, 

a;2 + 2xy -\- y^ + 4:X -{- 4y = {x -\- yY + 4 {x + y) = 117. 
lt\ now, in the equation 

{x-\-yy-{-4{x + y) = n7, 
we regard x -\- y as a single unknown quantity, we shall have 



x + y=-2zt^n7 + 4y 
hence, x-^y=— 2-{-ll=9. 

and x-\-y= —2 — U = —IS] 

whence, x = 9 — y, and a; = — 13 — y. 

Substituting these values of x in the second equation, we have 

y^ -}- 2y = 35, for a; — 9 — y, 
and y2 + 2y = 57, for a; = — 13 — y. 

The first equation gives, 

y =r 5, and y = — 7, 
and the second, 

y = - 1 +^58; and y = - I -^ ^58. 
The corresponding values of x, are 

X = 4:, X = 16', 

a: =-12-^58, and ar^.= - 12 +^58. 
4. Find the values of x and y, in the equations 
x^y^ + ^2/^ + ^y = 600 — (y + 2) ri'^y^ 
a: + y2 — 14 — y. 
From the first equation, we have 

a;2y2 + (3/2 + 2y) ar^yZ + ^y2 ^ xy = 600, 
or, a;V (1 + y^ + 2y) + ^y (1 + y) = 000, 

or, again, x^f (1 + yf + ^^y (1 + y) = COO ; 

12 



178 ELEMENTS OF ALGEBRA. [CHAP. VI. 

which is of the form of an equation of the second degree, re- 
garding x]j (1 -f y) as the Unknown quantity. Hence, 

zy (1 + 2/) = - i ±y600 + A = - 1 ±y^?^ ; 

and if we discuss only the roots which belong to the -f- value 
of the radical, we have 

24 
and hence, x — ; — r . 

y -\-y^ 

Substituting this value for x in the second equation, we have 

(2/2 + 2/?-14(2/^ + 2/) = -24; 
whence, 3/2 -|- y = 12, and y^ _^ ^ _ 2. 

From the first equation, we have 

y=-y±|- = 3, or -4; 

tmd the corresponding values of x^ from the equation 

24 

^--—-=^2. 

y^ -Vy 

From the second equation, we have 

y — \, and yr=— 2; 
which gives x — VI. 

5. Given, x^y + .'^y^ = 6, and x'^y'^ -f ^^y^ = 12, to find the 

'values of x and y. , C a; = 2 or 1, 

^ Ans. \ ' 

( y =: 1 or 2. 

/» /-I. ( rc^ _j_^ _|_ y _ 18 — y2 I to find the values of 

6. Given, -J ^ r 

( xy — K> ) X and y. 

' (y = 2, or 3; or - 3 =f y^ 

Pi'ohlems giving rise to Equations of the Second Degree con 
taining two or more unknown quantities. 

1. Find two numbers such, that the sum of the respectlv . 
products of the first multiplied by a, and the second multiplieo 
by 6, shall be equal to 2^ ; and the product of the one by 
the other equal to p. 



•CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 179 

Let X and y denote the required numbers, and yve have 
ax -{- hy =. 2s, 
and xy =p. 



From the first 



2s 
y = — 



b ' 
whence, by substituting in the second, and reducing, 

aa;2 — 2sx = — bp. 

g I . 

Therefore, ^ x z= — ± — ■/ s^ _ abp. 

a a V 

and consequently, y = — ^ yV ^^ ~ ^^^' 

Let a = 6 = 1 ; the values of a:, and ?/, then reduce to 
ic = s ± ^iP- — />, and ?/ = s =+: ^ iP- — 'p ; 

whence we see that, under this supposition, the two values 
of a: are equal to those of y, taken in an inverse order; which 
shows, that if 

s -\- .J^ — p represents the value of x^ 8 — J^ — p 

will represent the corresponding value of ?/, and conversely* 

This relation i^ explained by observing that, under the last 
supposition, 'h? / /en equations become 

a; -}- y =r 2s, and xy = p; 

and 'ae 'jOrAtica is then reduced to finding two numbers of which 
the sur,t is 2s, and their product p ; or in other words, to divide 
a number 2s, into two such parts, that their product may be equal 
to a given number p. 

2. To find four numbers, such that the sum of the first and 
fourth shall be equal to 2s, the sum of the second and third 
equal to 2s', the sum of their squares equal to 4c^, and the 
product of the first and fourth equal to Ihe product of the 
second and third. 



180 ELEME^^rS OF ALGEBRA. LCHAP. V], 

Let «j X, y, an»i z, denote the numbers, respectively. Then, 
firom the conditions of the problem, we shall have 

u -\- z = 2s 1st condition ; 

x-\-y =2s' 2d 

«H-^^ + y2 + ^2=4c2 3d 

uz = xi/ 4th " 

At first sight, it may appear difficult to find the values of 
the unknown quantities, but by the aid of an auxiliary unknown 
quantity, they are easily determined. 

Let p be the unkno^vn product of the 1st and 4th, or 2d 
and 3d ; we shall then have 



iu-\-z = 2s, ) 
\ uz=p, ) 



which give, 

■ z 



and 

' X 



ix+y=2s\) 



= 


s+./s^- 
s-l/s^- 


-Pi 
-P' 


= 


s' + ^- 


-p, 



which give, . 

y = s'—^s'^—p. 



Now, by substituting these values of w, x, y, z, in the third 
equation of the problem, it becomes 

(, 4 ^s-^-pY + {s -^^7)2 + (,r 4.y7rZ7)2 



^-{s' -/7^^pf=Ac^', 
and by developing and reducing, 

4s2 + 4s'2 — 4p = 4c2 ; hellce, p = s"^ -\- s"^ — c^. 

Substituting this value for p, in the expressions for u, x, y, 5, 
WQ find 

= s+^c^-s'\ ix = s'-\-^c^-s% 



iu = s+^c^ — s"^, ixz=s'-\-. 

z =s—Jc^ — a'\ }y = s'—. 



y^c2-a'2, iy = s'-^c-^ — s\ 

These values evidently satisfy the last equation of ihe 
problem; for 

uz z= {S +^c2-5'2) {S _yc2-s'2) ^ ^2 _ ^2 -f s'2, 



Xy = (s'-fyc2-s2) (s/ _y^;2^^2) = 5'2 - C2 + « «. 



CHAP, yi.] EQUATIONS OF THE SECOND DEGREE. 181 

Kemare. — This problem shows how much the introduction 
of an unknown auxiliary often facilitates the determination of 
the principal unknown quantities. There are other problems 
of the same kind, which lead to equations of a degree supe- 
rior to the second, and yet they may be resolved by the aid of 
equations of the first and second degrees, by introducing unknown 
auxiliaries. 

3. Given the sum of two numbers equal to a, and the sum 
of their cubes equal to c, to find the numbers 

{X -\- y ^=z a 
^■^ + 2/^ = c. 
Putting X =z s -\- z^ and y — s — z^ we have a z=2s^ 

i x^ = s^ -\- Ss^z -\- Ssz'^ -\- z^ 
and •< 



hence, by addition, x^ -^y^ — 2s'^ -f- Qs^ 



whence, z^ == — , and 2 = =t\ /- 

05 V 



c - 2s''^ 



, /c-26-3 ^ /C - 26-3 

and by substituting for s its value, • 



a , //c — ^a^\ a h 

2V(-3f-)-2V- 



4c 

x = 



12a 



, a lie — \<Ji\ a I ^c — a^ 

and 3, = _^^(_^j = _^^___. 

4, The sum of the squares of two numbers is expressed by 
«, and the difference of their squares by b : what are the 
numbers? A^^T^ fT^l 

^"^•V-2-' V-2- 



5. What three numbers are they, which, multiplied two and 
two, and each product divided by the third number, give th€ 
quotients, a, 6, c? 



182 ELEMENTS OF ALaEBRA. |CHAP. VI. 

6. The sum of t^YO numbers is 8, and the sum of their 
eu"bes is 152: what are the numbers'? Ans. 3 and 5. 

7. Find two numbers, whose difference added to the differ- 
ence of their squares is 150, and whose sum added to the 
sum of their squares, is 330. Ans. 9 and 15. 

8. There are two numbers whose difference is 15, and half 
theii product is equal to the cube of the lesser number : what 
are the numbers 1 Ans. 3 and 18. 

9. What two numbers are those whose sum multiplied by 
the greater, is equal to 77 ; and whose difference, multiplied 
bj the lesser, is equal to 12 1 

Ans. 4 and 7, or | V2 and ^J V2! 

10. Divide 100 into two such parts, that the sum of their 
square roots may be 14. Ans. 64 and 36. 

11. It is required to divide the number 24 into two such 
parts, that their product may be equal to 35 times Iheir differ- 
ence. Ans. 10 and 14. 

12. What two numbers are they, whose product is 255, and 
the sum of whose squares is 514 1 Ans. 15 and 17. 

13. There is a number expressed by two digits, which, when 
divided by the sum of the digits, gives a quotient greater by 
2 than the first digit ; but if the digits be inverted, and the 
resulting number be divided by a number greater by 1 than 
the sum of the digits, the quotient will exceed the former 
quotient by 2 : what is the number 1 Ans. 24. 

14. A regiment, in garrison, consisting of a certain number of 
companies, 'receives orders to send 216 men on duty, each com- 
pany to furnish an equal number. Before the order was exe- 
cuted, three of the companies were sent on another service, 
and it was then found that each company that remained would 
have to send 12 men additional, in order to make up the com- 
plement, 216. How many companies were in the regiment, and 
what number of men did each of the remaining companies send 

Ans. 9 companies : each that remained sent 36 men. 



CHAP VI.j EQUATIONS OF THE SECOND DEGKEE. 183 

15. Find three numbers such, that their sum shall be 14, the 
sum of their squares equal to 84, and the product of the first 
and third equal to the square of the second. , 

Ans. 2, 4 /'and 8. 

16. It is required *to find a number, expressed by three 
digits, such, that the sum of the squares of the digits shall 
be 104; the square of the middle digit to exceed twice the 
product of the other two by 4 ; and if 594 be subtracted from 
the number, the remainder will be expressed by the same 
figures, but with the extreme digits reversed. Ans. 862. 

17. A person has three kinds of goods which togetner cost $230/^. 
A pound of each article costs as many -J^ dollars as there are 
pounds ill that article : he has one-third more of the second than of 
the first, and 3^ times as much of the third as of the second: How 
many pounds has he of each article ? 

An}^. 15 of the 1st, 20 of the 2d, 70 of the 3d. 

18. Two merchants each sold the same kind of stuff: the 
second sold 3 yards more of it than the first, and together, 
they received 35 dollars. The first said to the second, " I 
would have received 24 dollars for your stufi"." The other r&. 
plied, " And I would have received 12J dollars for yours." 
How many yards did each of them sell'? 



( 1st merchant 15) (5 



19. A widow possessed 13000 dollars, ^vhich she divided into 
two parts, and placed them at interest, in such a manner, that 
the incomes from them were equal. If she had put out the first 
portion at the same rate as the second, she would have drawn 
for this part 360 dollars interest; and if she had placed the 
second out at the same rate as the first, she would have drawn 
for it 490 dollars interest. What were the twc rates of interest'! 

Ans. 7 and 6 per cent. 



CHAPTER Vn. 

PORMATIOI' OP POWERS BINOMIAL THEOREM — EXTUACriON OF HOOT* OP 

ANY DEGREE OF RADICALS. 

128. The solution of equations of the second degree supposes 
the process for extracting the square root to be known. In 
like manner, the solution of equations of the third, fourth, &c., 
degrees, requires that we should know how to extract the third, 
fourth, &c., roots of any numerical or algebraic quantity. 

The power of a number can be obtained by the rules for 
multiplication, and this power is subject to a certain law of for- 
mation^ which it is necessary to know, in order to deduce the 
root from the power. 

Now, the law of formation of the square of a numerical or 
algebraic quantity, is deduced from the expression for the squar** 
of a binomial (Art. 47) ; so likew^ise, the law of a power of 
any degree, is deduced from the expression for the same power 
of a binomial. We shall therefore first determine the law for 
the formation of any power of a binomial. 

129. By taking the binomial x -\- a several times, as a factor, 
the following results are obtained, by the rule for multiplicatiou • 

(x -\- a) z=z X -f «, 
\x + af = a:2 -L 2ax + a^, 
{x. + af = .r3 + 3arc2 -f- Za^x -J- a^, 
(ri -I- ay = x^ + 4ax^ + 6a^x^ + 4a^x + a\ 
(x + ay = x^-h 5ax* + lOa^x^ + lOa^x^ + Sa^a: -f «*. 
By examining these powers of x + «, we readily discover th€ 
law according to which the exponents of the powers of a de 



CHAP. VII.J PERMUTATIONS AND COMBINATIONS. 185 

crease, and those of the powers of a increase, in the successive 
terms. It is not, however, so easy to discover a law for the 
formation of the co-efficients. Newton discovered one, by means 
of which a binomial may be raised to any power, without per 
forming the multiplications. He did not, however, explain the 
course of reasoning which led him to the discovery ; but the law 
has since been demonstrated in a rigorous manner. Of all the 
known demonstrations of it. the most elementary is that which 
is founded upon the theory of combinations. However, as the 
demonstration is rather complicated, we will, in order to simplify 
it, begin by demonstrating some propositions relative to permu- 
tations aiid combinations, on which the demonstration of the 
binomial theorem depends. 

Of Permutations^ Arrangements and Oomhinations. 

130» Let it be proposed to determine the whole number of 
ways in which several letters, a, 6, c, c?, &c., can be written, 
one after the other. The result corresponding to each change 
in the position of any one of these letters, is called a 2'^er 
mutation. 

Thus, the two letters a and b furnish the two permutations^ 

sLb and ba. 

rcah 

acb 
' In like manner, the three letters, a, 5, c, furnish abc 

six permutations. | cha 

boa 

.bai 

Permutations, are the results obtained by writing a certain 

number of letters one after the other^ in every possible order^ in 

such a manner that all the letters shall enter into each result^ and 

each letter enter but once. 

To determine the number of permutations of which n letters are 
susceptible. 

Two letters, a and 6, evidently give two per- j ab 

mutations. \ ba 



186 ELEMENTS OF ALGEBRA. [CHAP. VL 

Therefore, the number of permutations of tM^o letters is ex 
pressed by 1x2. 

Take the three letters, a, 6, and c. Reserve r c 

either of the letters, as c, and permute the other -j ai 

t^'o, giving ' ba 



''cab 
acb 
abc 
cba 
bca 



Now, the third letter c may be placed before a6, 
between a and 6, and at the right of ab ; and the 
same for ba : that is, in one of the first 'permuta- 
tions^ the reserved letter c may have three different 
places^ giving three permutations. And, as the same 
may be shown for each one of the first permutations, 
it follows that the whole number of permutations of 
three letters will be expressed by, 1x2x3. [ bac 

If, now, a fourth letter d be introduced, it can have four 
places in each one of the six permutations of three letters : 
hence, the number of permutations of four letters will be ex- 
pressed by, 1 X 2 X 3 X 4. 

In general, let there be n letters, a, 5, c, &c., and suppose 
the total number of permutations of 7j — 1 letters to be. known; 
and let Q denote that number. Now, in each one of the Q per- 
mutations, the reserved letter may have n places, giving n per- 
mutations : hence, when it is so placed in all of them, the 
entire number of permutations will be expressed b^/ Q X n. 

If ?2, = 5, Q will denote the number of permutations of four 
quantities, or will be equal to 1x2x3x4; hence, the num. 
ber of permutations of five quantities will be expressed by 
1x2x3x4x5. 

If n — 6, we shall have for the number of permutations of 
six quantities, 1x2x3x4x5x6, and so on. 

Hence, if Y denote the number of permutations of n letters, 
w^ shall have 

F=^X^ = 1. 2. 3. 4. . . . {n'-V)n\ that is, 

The number of per7nutations of n letters^ is equal to the con,' 
t^iucd product of the n.itural numbers from 1 to n inclusively. 



CHAP. VI.] PERMUTATIONS AND COMBINATIONS. 187 

Arrangements. 

131, Suppose we have a number m, of letters a, 6, c, d, &c. 
\i they are written in sets of 2 and 2, or 3 and 3, or 4 and 4 
... in every possible order in each set, such results are called 
arrangements. 

Thus, ab^ ac, ad, . . . ha, he, hd, . . . ca, ch, cd, . . . are ar- 
••angements of m letters taken 2 and 2 ; or in sets of 2 each. 

In like manner, ahc, ahd, . . . bac, bad, . . . acb, acd, . . . are 
irrangements taken in sets of 3. 

Arrangements, are the results obtained by writing a number m 
f letters, in sets of 2 and 2, 3 and 3, 4 and 4, . . . n and n ; 
v\e letters in each set havifig evejy possible order, and m being 
always greater than n. 

If we suppose m = n, the arrangements, ^^ken n and n, be- 
come permutations. 

Having given a number m of letters a, b, c, d, . . . to deter- 
mine ihe total number of arrangements that may he formed of them 
hy taking them n in a set. 

Let it be proposed, in the first place, to arrange three letters, 
a, b and c, in sets of two each. 

First, arrange the letters in sets of one each, and 
for each set so formed, there will be two letters 
reserved: the reserved letters for either arrange- 
ment, being those which do not enter it. Thus, with 
reference to a, the reserved letters are h and c ; with reference 
to b, the reserved letters are a and c; and with reference to c, 
they are a and b. 

Now, to any one of the letters, as a, annex, in ^" 

succession, the reserved letters b and c : to the 

second arrangement b. annex the reserved letters a •{ , 

. he 

and c; and to the third arrangement, c, annex the 

reserved letters a and b. 

Since each of the first arrangements gives as many new 

arrangements as there are reser ved letters, it follows, thai the 



ca 
ch 





'ab 




ae 




ad 




ha 


1 


he 
bd 
c a 

cb 




cd 




da 




dh 




dc 



188 ELEMENTS OF ALGEBRA, [CHAP. VIL 

number of arrangements of three letters taken^ two in a set, will be 
equal to the number of arrangements of the same letters taken ont 
in a set, multiplied by the number of reserved letters. 

Let it be required to form the arrangement of four letters, 
a, 5, c and c?, taken three in a set. 

First, arrange the four letters in sets of two : there 
will then be for each arrangement, two reserved let- 
ters. Take one of the sets and write after it, in suc- 
cession, each of the reserved letters : we shall thus 
form as many sets of three letters each as there are 
reserved letters ; and these sets differ from each other 
by at least the last letter. Take another of the first 
arrangements, and annex, in succession, the reserved 
letters ; we shall again form as many different arrange- 
ments as there are reserved letters. Do the same for. 
all of the first arrangements, and it is plain, that the 
whole number of arrangements which will be formed, of four 
letters, taken 3 and 3, will be equal to the iiumber of arrange- 
ments of the same letters, taken two in a set, multiplied by the 
number of reserved letters. 

In general, suppose the total number of arrangements of m 
letters, taken n — 1 in a set, to be knowTi, and denote this num- 
ber by F. 

Take any one of these arrangements, and annex to it, in suc- 
cession, each of the reserved letters, of which the number i? 
m — {n — 1), or m — n -\- \. It is evident, that we shall thus 
form a number m — n -\- \ of new arrangements of n letters, 
each diffcilng from the others by the last letter. 

Now, take another of the first arrang«ments of n — 1 letters, 
and annex to it, in succession, each of the m — n -\- \ letters 
which do not enter it ; we again obtain a number m — n -{- \ of 
arrangements of n letters, differing from each other, and' from 
those obtained as above, by at least one of the n — \ first letters. 
Now, as we may in the same manner, take all the P arrange- 
ments of the m letters, taken »i — 1 in a set, and annex to thera 



CHAP, VII.] PERMUTATIONS AND COMBINATIONS. 189 

in succession, each of the m — n -{- 1 other letters, it follows 
that the total number of arrangements of m letters, taken n in 
a set, is expressed by 

F{m — n-i-l). 

To apply this, in determining the number of arrangements of 
m letters, taken 2 and 2, 3 and 3, 4 and 4, or 5 and 5 in a 
set, make n =z 2 ', whence, m — n -\- \ zzz m — 1; P in this 
case, will express the total number of arrangements, taken 2 — 1 
and 2 — 1, or 1 and 1 ; and is consequently equal to m; there- 
fore, the expression 

P{m — n -\- \) becomes m{m — 1). 

Let 71 = 3 ; whence, m — n -\-\ ■= m — 2; P will then ex- 
press the number of arrangements taken 2 and 2, and is equal 
to 77i(m — 1) ; therefore, the expression becomes 

m{m — \){m — 2). 

Again, take 7^ = 4 : whence, m — w + l=:m — 3: P will ex 
press the number of arrangements taken 3 and 3, and therefore 
the expression becomes 

m[m — 1) (m — 2) (m — 3), and so on. 

Hence, if we denote the number of arrangements of m let- 
ters, taken ?2. in a set by X, we shall have, 

X = P{m — n -{- 1) = m (ni — 1) (»i — 2) . . (m — n -\- 1) -, that is, 

The number of arrangements of jn letters^ taken n in a set^ is 
equal to the continued product of the natural numbers from m 
down to m — n -\- \^ inclusively. 

If in the preceding formula m be made equal to ti, the ar 
rangements become permutations, and the formula reduces to 

X=n{n- l)(?i-2) . . . . 2 . 1; 

or, by reversing the order of the factors, and writing Y for X^ 

F=l . 2 . 3 .... (71- 1)71 ; 

the same formula as deduced in the last artiole. 



190 ELEMENTS OF ALGEBRA. [CHAP. VU 

Combinations. 

132» When the letters are disposed, as in the arrangements, 
2 and 2, 3 and 3, 4 and 4, &c., and it is required that any 
two of the results, thus formed, shall differ by at least one 
letter, the products of the letters will be different. In this case, 
the results are called combinations. 

Thus s(6, ac, be, . . . ad, bd, . . . are combinations of the let- 
ters a, &, c, and d, &;c., taken 2 and 2. 

In like manner, abc, abd, . . . acd, bed, . . . are combinations 
of the letters taken 3 and 3 : hence, 

Combinations, are arrangements in which any two loill differ 
from each other by at least one of the letters which enter them. 

To determine the total number of different coinbinations thai 
can be formed of m letters, taken n in a set. 

Let X denote the total number of arrangements that can be 
formed of m letters, taken n and n : Y the number of per 
mutations of n letters, and Z the total number of different 
combinations taken n and 7i. 

It is evident, that all the possible arrangements of m letters 
taken n in a set, can be obtained, by subjecting the 7i letters 
of each of the Z combinations, to all the permutations of which 
these leitters are susceptible. Now, a single combination of n 
letters gives, by hypothesis, Y permutations or arrangements • 
therefore Z combinations will give Y X Z arrangements ; and 
as X denotes the total number of arrangements, it follows that 





X=Y 


XZ; 


whence 


, Z = 


X 
Y' 






But we 


have (Art. 


130), 














r=Q 


Xn=: 


1.2. 


3 . . . 


. n, 






and (Art. 1 


131), 














X=F(m 


,-» + !) = 


~ m {in 


-l)(m 


-2). 


. . . 


{m — 


• n^l 


therefore 


} 
















2 x?i 


m(m - 


- 1) (m - 


-2) . . 


. . 


[m — 


n-\-l) 




1.2. 


3 . . . 


, , 


. . n 





that is. 



CHAP. VII. J BINOMIAL THEOREM. 191 

The number of combinations of m letters taken n in a set, 
is equal to the continued product of the natural numbers from 
m down to m — 7i -f- 1 inclusively^ divided by the continued 
'product of the natural numbers from \ to n inclusively. 

133. If Z denote the number 'of combinations of the m let- 
ters taken w in a set, we have just seen that 
m{m — \){m — 2)....{m — n+l) 
^"^ 1.2.3 n ^^^- 

If Z' denote the number of combinations of m letters taken 
{m — n) in a set, we can find an expression for Z' by chang- 
ing n into 'm — n in the second member of the above formula ; 
whence 

_ m(m-l)(m-2) (n + I) 

1.2.3...... {m-n) ^ ^* 

If, now, we divide equation (1) by (2), member by member, 
and arrange the factors of both terms of the quotient, we 
shall have 

Z _ 1 . 2 . 3 . . . . (m — to) X (m — ?z 4- 1) . . . {m — V)m 
~Z' " 1.2.3.... . 9^ X (n + 1) . ' (m — l)m' 

The numerator and denominator of the second member are 
equal to each other, since each contains the factors, 1, 2, 3, 
&c., to m; hence, 

— = 1, or Z =: Z' ', therefore, 

Li 

The number of combinations of in letters^ taken n in a set, is 
equal to the number of combinations of m letters, taken m ^ n in 
a set. 

Binomial Theorem. 

134. The object of this theorem is to show how to ficd any 
power of a binomial, without going through the process of con 
tmued multiplication. 

135. The algebraic equation which indicates the law of for- 
mation of any power of a biromial, is called the Binomiat 
Formula. 



192 



ELEMENTS OF ALGEBRA. 



[CHAP. Vll. 



In order to discover this law for the mth power of the bino- 
mial X -r-a, let us observe the law for the formation of the 
product of several binomial factors, x -i- a, x -\- b, x -\- c, x -\-d 

. . of which the first term is the same in all, and the second 



terms different 


X -\- a 






X -\- b 


- 


1st product 


- x^ -\- a 
X -\- c 


X -\- ab 


2d ■ . . . 


, x^ -\- a 


x^ 4- ah 




^b 


+ ac 




+ c 


^bc 




X ^d 





+ abc 



Sd 



re* + a 


x^ 4- ah 


x^ -\- abc 


-\-b 


+ ac 


4- ahd 


+ c 


4- ad 


4- acd 


-\-d 


+ be 
+ bd 
+ cd 


4- bed 



X 4- cJbcd 



These products, obtained by the common rule for algebraic 
multiplication, indicate the following laws : — 

1st. With respect to the exponents, we observe that the ex- 
ponent of a:, in the first term, is equal to the number of bino- 
mial factors employed. In each of the following terms to the 
right, this exponent is diminished by 1 to the last term, where 
it is 0. 

2d. With respect to the co-efficients of the different powers 
of a;, that of the first term is 1 ; the co-efficient of the second 
term is equal to the sum of the second terms of the binomials; 
the co-efficient of the third term is equal to the sum of the 
products of the different second terms, taken two and two; 



CHAP. VII.] BINOMIAL THEOREM. 193 

the co-efficient of the fourth term is equal to the sum of their 
different products, taken three and three. 

Reasoning from analogy, we might conclude that, in the pr(v 
duct of any number of binomial factors, the co-efficient of the 
term which has n terms before it, is equal to the sum of the 
different products of the second terms of the binomials, taken 
n and n. The last term of the product is equal to the con- 
tinued product of the second terms of the binomials. 

In order to prove that this law of formation is general, sup- 
por>e that it has been proved true for the product of m bino- 
mials. Let us see if it will continue to be true when the 
product is multiplied by a new binomial factor of the same 
form. 

For this purpose, suppose 

to be the product of 7?i binomial factors; Nx'^~^ repiesenting the 
Cferm which has n terms before it, and Mx^—''^-^^ the term which 
immediately precedes. 

Let X -\- k be the new binomial factor by which wo multiply ; 
the product, when arranged according to the powers of «, 
will be 



^ k\ -{- Ak 









from which we perceive that the law of the exponents is evi- 
dently the same. 
^, With respect to the co-efficients, we observe; 
W 1st. That the co-efficient of the first term is 1 ; and 

2d. That A -\- k, or the co-efficient of a:"', is the sum of Hie 
second terms of the m -|- 1 binomials. 

3d. Since, by hypothesis, B is the sum of the different products 
of the second terms of the m binomials, taken two and two, and 
since A x k expresses the sum of the products of each of the 
second terms of the first m binomials by the new second term k ; 
therefore, B -\- Ak is the sum of the different products of Uve 
\ second terms of the m-\-\ binomials, taken two and two, 

13 



194 ELEMENTS OF ALGEBRA. [CHAP. VIT. 

Id general, since iV expresses the sjm of the products of the 
second terms of the m binomials, taken n and n, and M the sum 
of their products, taken n — \ and « — 1, therefore N -^ Mk^ 
or the co-efficient of the term^ which has n terms before it, will be 
equal to the sum of the diiFerent products of the second terms 
of the m + 1 •binomials, taken n and n. The last term is 
equal to the continued product of the second terms of the m-Y\ 
binomials. 

Hence, the law of composition, supposed true for a number m 
of binomial factors, is also true for a number denoted by m -J- 1. 

But we have shown the law of composition for 4 factors, 
hence, the same law is true for 5 ; and bemg true for 5, it 
must be for 6, and so on; hence, it is general. 

136. Let us take the equation, 

[x ^ a){x -\- h){x ^ c) . ... = a;"» + Ax"^^ + Bx"^"^ .... 

+ Nx^^ . . . . + tf; 

containing in the first member, m binomial factors. If we make 

a ^=^h ^1^ c :=: d . . . . &c., 
the first member becomes, 

{x + a)'^. 
In the second member the co-efficient of x^ will still be 1. 
The co-efficient of a:'"-^, being a -\- b -\- c -\- d^ . . . will become 
a taken m times ; that is, ma. 
The co-efficient of a;"^^^ being 

ab -\- ac -\- ad . . . . reduces to a^ -j- a^ -f a^ . . , 
that is, it becomes a^ taken as many times as there are com 
binations of m letters, taken two and two, and hence reduces 
(Art. 132), to 



m — 1 , 
m . — - — a 



2 

The oo-efficient of x'^^ reduces to the product of a^, multi 
plied by the number of different combinations of m letters, 
taken three and three ; that is, to 
m — 1 m — 2 



2 • 3 ' 



\ <fec 



CHAP. VII] BINOMIAL THEOREM. 195 

Let us denote the general term, that is ^ne one Avhich haa 
n terms before it, by iVlc"*-". 

Then, the eo-efficient iV will denote the sum of the prod'icts 
of the second terms, taken n and n ; and when all the 
second terms are siipposed equal, it becomes equal to a" mul- 
tiplied by the number of combinations of m letters, taken 
n and n. Therefore, the co-efficient of the general term ^Art. 

132), is 

P(m — ?^ + l) 

Qxn "^ ' 
hence, we have, by making these substitutions, 

(x -f a)^ = x"^ + max"^-^ + m , — - — a2^OT-2 

t 

m — Im — 2, „ , F(m — n-^1) 
4- m . — -—. — - — a^x"^^ . . . + ^ ^ — - a«:c"»-'* . , . -f- a* 

which is the binomial formula. 
The term 

__v ) a«a;'^" 

Qn 

is called the general term, because by making n = 2, 3, 4, (fee, 

all the others can be deduced from it. The term which im 

mediately precedes it, is 

F P 

-— a"-'a;"'-" + \ since — - 

evidently expresses the number of combinations of m letters 

taken n — \ and n — 1. Hence, we see, that 
. P(m-n+\) 

^ . Qx7i ' 

which is called the nuraeiical co-efficient of the general term, 

P 

Is equal to the numerical co-efficient -— - of the preceding term, 

multiplied by m — n -j- 1, the exponent of x in that term, and 
divided by ?i, the number of terms preceding the required term, 

llie simple law, demonstrated above, enables us to determine 
the numerical co-efficient of any term from that of the preceding 
term, by means of the following 



L^ ELEMENTS OF ALGEBRA. [CHAP. VIL 

RULE. 

The numerical co-efficient of any term after the first, is formed 
&y multiplying that of the preceding term by the exponent of 
9 in that term^ and dividing the product by the number of 
Urms which precede the required term. 

137. Let it be required to develop 

{x + ay. 

By applying the foregoing principles, we find, 

(x }- ay =zx^ + 6ax^ -f 16a'^x^ + 20a^x^ + 15a*a;2 + 6a^x + a*. 

Having written the fir-st term x^, and the literal parts of the 
other terms, we find the numerical co-efficient of the second 
term by multiplying 1, the numerical co-efficient of the first 
term, by 6, the exponent of x in that term, and dividing by 
1, the number of terms preceding the required term. To obtain 
the co-efficient of the third term, multiply 6 by 5 and divide 
the product by 2 ; we get 15 for the required number. The 
other numerical co-efficients may be found in the same manner 

In like m_anner, we find 

{x + a)^o =z a;io -f lOax^ -f 4:Da^x^ + 120aV -f 2l0a^x^ 
+ 2b2a^x^ -\- 210a^x* -f 120a^;r3 _|_ 45^8^2 _|_ iq^^x + a^^ 

138t The operation of finding the numerical co-efficients may 
be much simplified by the* aid of the following principle. 

We have seen that the development of {x -f a)'", contains 
•» 4- 1 terms ; consequently, the term which has n terms after 
it, has m — n terms before it. Now, the numerical co-efficient 
of the term which has n terms before it is equal to the num- 
ber of combinations of m letters taken n in a set, and the 
numerical co-efficient of that term which has n terms after it, 
or m — n before it, is equal to the number of combinations of 
m letters taken m — n in a set; but we have shown (Art. 133) 
that these numl/ers are equal. Hence, 

In the development of any power of a binomial of the form 
Ix -f- af, the numerical co-efficients of terms at equal distances from 
the two extremes^ are equal to each other. 



CHAP. VII.] BINOMIAL THEOREM. 197 

We see that this is the case in both of the examples above 
given. In finding the development of any power of a binomial, 
we need find but half, or one more than half, of the numerical 
co-efficients, since the remaining ones may be written directly 
from those already found. 

139. It frequently happens that the terms of the binomial, 
to which the formula is to be applied, contain co-efTicienta 
and exponents, as in the following example. 

Let it be required to raise the binomial 
3a2c — 2hd 
to the fourth' power. 

Placing oa?c = x and — 2bd =r y, , we have 

(x -^ yY z= x^ -\- 4x^y + Qx'^y^ + 4:xy^ -\- y^ ; 
and substituting for x and y their values, we have 
(3a2c - 2bdY = (3a2c)* + 4 {Sa'^cf ( - 2bd) -f 6 {Sa^cY (— 2bdY 

+ 4 {Sa^c) (- 2bdy + (- 2bdY, 
ocj^ by performing the operations indicated, 
"""'{Ba^c — 2bdY = 81aM - 21Qa^c^d + 21QaHWd^ - 96ahb^d^ 

+ 16b^d\ 
. " The terms of the development are alternately plus and 
I fiainus, as they should be, since the second term is — . 

140. A power of any polynomial may easily be found by 
means of the binomial formula, as in the following example. 

Let it be required to find the third power of 

a -\- b -\- c. 
First, put b -}- c = d. 

Then {a-^b+cy = {a -{- df = a^ ^ Sa'^d + Sad^ 4- d^, 
and by substituting for the value of d, 

(a 4- 6 H- c)3 = a3 + Sa% + Sab^ + 6' 

3a2c + Sb^c -h Qabc 
-f 3ac2 4- 36c2 



198 ELEMENTS OF ALGEBRA. [CHAP. VII. 

This development is composed of tlie sum of the cubes of tht 
three teivns, plus the sum of the results obtained by multiplying 
three times the square of each terra, by each of the other terms in 
succession, plus six times the product of the three terms. 

To applj the preceding formula to the development of the 
cu"be of a trinomial, in which the terms are affected with co- 
efTjcierts and exponents, designate each term by a single letter^ 
and perform the operations indicated ; then replace the letters 
introduced, by their values. 

From this rule, we find that 

(2a2 _ Aab + 362)3 ^ g^e _ 49^55 -j- \S2a^b^ — 208a^b^ 
+ 198a26* - lOSab^ + 2766. 
The fourth, fifth, &c., powers of any polynomial can be de- 
veloped in a similar manner. 

Extraction of the Cube Boot of Numbers. 

141. The cube root of a number, is such a number as being 
taken three times as a factor, will produce the given number. 

A number whose cube root can be exactly found, is called a 
perfect cube ; all other numbers are imperfect cubes. 

The first ten numbers are, 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10; 

and their cubes, 
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. 

Conversely, the numbers in the first line are the cube roots 
of the corresponding numbers in the second. 

If we wish to find the cube root of any number less than 
1000, we look for the number in the second line, and if 't is 
there written, the corresponding num.ber in the first line will be 
its cube root. If the number is not there written, it will fiili 
between two numbers in the second line, and its cube root 
will fall between the corresponding numbers in the first line. 
In this case the cube root cannot be expressed in exact parts 
of 1 ; hence, the given num ber must be an imperfect cube (Re- 
mark III, Art. 95). 



CHAP. Vir.j C JBE ROOT OF NUMBERS. 199 

If the given number is greater than 1000, its cube root will 
be greater than 10 ; that is, it will contain a certain number 
of tens and a certain number of units. 

Let us designate any number by i\r, and denote its tens by 
a, and its units by b ; we shall have, 

N^.a-\-b', whence, N^ = a^ + ^cfib + Zab'^ + b^ ', that is, 

The cube of a number is equal to the cube of the tens, plus three 
times the product of the square of the tens by the units, plus three 
times the product of the tens by the square of the units, plus the 
cube of the units. 
Thus (47)3.=: (40y + 3 X (40)2 X 7 + s x40 X (7)^ + (7)3 = 103823. 

Let us now reverse the operation, and find the cube root of 
103823. 

103 823 
64 



42 X 3 = 48 I 398'23 



47 


48 


47 


~8 


48 


47 




384 


329 




192 
2304 


188 




2209 




48 


47 




18432 


15463 


1 


9216 


8836 



110592 103823 

Since the number is greater than 1000, its root vfill contain 
tens and units. We will first find the number of tens in the 
root. Now the cube of tens, giving at least thousands, we point 
off three places of figures on the right, and the cube of the num- 
ber of tens will be found in the number 103, to the left of this 
peiiod. 

The cube root of the greatest cube contained in 103 being 4, 
this is the number of tens in the required root. Indeed, 103823 
is evidently comprised between (40)- or 64,000, and (50)^ or 
125,000 ; heuce, the required root is comprised between 4 tens 
and 5 tens : that is, it is composed of 4 tens, plus a certain 
number of units less than ten. 



200 ELEMENTS OF ALGEBRA. LCHAP. VII. 

Having found the number of tens, subtract its cube, 64, from 
103, and there remains 39, to which bring down the part 823ij 
and we have 39823, which contains three times the product of 
the square of the tens by the units^ plus three times the product 
of the tens by the square of the units, plus the cube of the units. 

Now, as the square of tens gives at least hundreds, it follows 
that the product of three times the square of the tens by the 
units, must be found in the part 398, to the left of 23, which 
is separated from it by a dash. Therefore, dividing 398 by 48, 
which is three times the square of the tens, the quotient 8 will 
be the units of the root, or something greater, since 398 is 
composed of three times the square of the tens by the units, and 
generally contains numbers coming from the tv»^o other parts. 

We may ascertain whether the figure 8 is too great, by form- 
ing from the 4 tens and 8 units, the three parts which enter into 
39823 ; but it is nmch easier to cube 48, as has beeM done in 
the above table. Now, the cube of 48 is 110592, which is 
greater than 103823; therefore, 8 is too great. By cabhig 47, 
we obtain 103823 ; hence the proposed number is a ]-erfeot cube, 
and 47 is its cube root. 

By a course of reasoning entirely analogous to that pi:ri^^ucd 
in treating of the extraction of the square root, w^e nay shew 
that, when the given number is expressed by more than six 
figures, we must point off the number into periods of throe figui «s 
each, commencing at the right. Hence, for the extraction of the 
cube root of numbers, we have the following 

RULE. 

I. Separate the given number into periods of three figures each, 
beginning at the right hand; the left hand period will often con 
tain less than three places of figures. 

II. Seek the greatest perfect cube in the first 2^6riod, on the left, 
and set its root on the right, after the manner of a quotient ip 
division. Subtract the cube of this number from the first period^ 
and to the remainder bring down the first fygure of the next period^ 
and call this number the dividend. 



(!HAP. VII.J EXTRACTION OF ROOTS. 201 

III. Take three times the square of the root just found for a 
divisor, and see how often it is contained in the dividend, and 
place the quotient for a seomd figure of the root. Then cube the 
number thus found, and if its cube be greater than the first two 
periods of the given number, diminish the last figure by 1 ; but 
if it be less, subtract it from the first two periods, and to the 
remainder bring down the first figure of the next period, for a new 
dividend. 

IV. Take three times the square of the whole root for a new 
divisor, and seek how often it is contained in the new dividend ; 
the quotient will be the third figure of the root. Cube the number 
thus found, and subtract the result from the first three periods 
of the given number, and proceed in a similar way for all the 
periods. 

If there is no remainder, the number is a perfect cube, and the 
root is exact : if there is a remainder, the number is an inqyer- 
feet cube, and the root is exact to within less than 1. 

EXAMPLES. 

1. 3^48228544 Ans. 304. 

2. 3^27054036008 Ans. 3002. 

3. 3^483249 Ans. 78, with a remainder 8697. 

4. y^9T632508641 Ans. 4508, with a remainder 20644129. 

5. 3^32977340218432 Ans. 32068. 

^traction of the N*^ Root of Numbers. 

142* The n*^ root of a number is such a number as being 
taken n times as a flictor will produce the given number, n being 
my positive whole number. When such a root can be exactly 
found, the given number is a perfect n^^ power; all other num- 
bers are imperfect n^^ powers. 

Let N denote any number whatever. If it is expressed by 
less than n -\- \ figures, and is a perfect n'^' power, its n^^ root 
will be expressed by a single figure, and may be found by 



202 ELEMENTS OF ALGEBRA. LCHAP. VXL 

means cf a tab\3 containing the n^^ powers of the first ten 
uun)l>ers. 

If the number is not a perfect- ??.*'' power, it will fall between 
two n^^ powers in the table, and its root will fall between the 
n'^^ roots of these powers. 

If the given number is expressed by more than n figures, 
its root will consist of a certain number of tens and a certain 
number of units. If we designate the tens of the root by a, 
and the units by b, we shall have, by the binomial formula, 

n — \ 

N — {a -{- by = a"" -\- na''~'^b -f- n — ^— a^'-W +, &c. ; 

that is, the proposed number is equal to the . n^^ power of the 
tens, plus n times the p)rodiLct of the ii — 1*^ power of the tens 
by the units, plus other parts which it is not necessary to 
consider. 

Now, as the n^^ power of the tens, cannot be less than 
1 followed by n ciphers, the last n figures on the right, cannot 
make a part of it. They must then be pointed off, and the n** 
root of the greatest n^^ power in the number on the left will 
be the number of tens of the required root. 

Subtract the n^^ power of the number of tens from the num 
ber on the left, and to the remainder bring down one figure of 
the next period on the right. If we consider the num.ber thus 
fuund as a dividend, and take n times the {n — 1)^^ power 
of the number of tens, as a divisor, the quotient will evidently 
be the number of units, or a greater number. 

If the part on the left should contain more than ?i figures, the 
n figures on the right of it, must be separated from the 'rest, 
and the root of the greatest n*^ power contained in the part 
on the left extracted, and so on. Hence the following 

HULE. 

I. Separate the number N into periods of n figures each, he 
ginning at the right hand; extract the n^^ root of the greatest 
perfect 71*^ power contained in the hft hand period, it will be th& 
first figure of the root. 



CHAP. VII.] EXTRACTION OF ROOTS. 208 

II. Subtract this n^^ power from the left hand period and hrin^ 
down to the right of the remainder the first figure of the next 
period^ and call this the dividend. 

[11. Form the n—\ power of the first figure of the root, mul- 
tiply it by n, and see how often the product is contained in the 
dividend : the quotient will be the second figure of the root, or 
something greater. 

IV. Raise the number thus formed to the n^^ poiver^ then sub- 
tract this result from the two left-hand periods^ and to the new 
remainder bring down the first figure of the next period : then 
divide the .number thus formed by n times the n — 1 power of 
the two figures of the root already found, and continue this opera- 
tion until all the periods are brought down. 

EXAMPLES. 

1. What is the fourth root of 531441 % 

53 1441 I 27 
2*= 16_ 

4 X 23 r= 32 I 371 
(27)* = 531441. 

We first point off, from the right hand, the period of four 
figures, and then find the greatest fourth root contained in 53, 
the first period to the left, which is 2. We next subtract the 
4th power of 2, which is 16, from 53, and to the remainder 
37 we bring down the first figure of the next period. We 
then divide 371 by 4 times the cube of 2, which gives 11 for 
a quotient : but this we know is too large. By trying the num- 
bers 9 and 8, we find them also too large : then trying 7, we 
find the exact root to be 27. 

143» When the index of the root to be extracted is a multiple 
of two or more numbers, as 4, 6, . . . &c., the root can be ob- 
tained by extracting roots of more simple degrees, successively. To 
explain this, w^e will remark that, 

(a3)4 _ ^3 X a^ X 6/3 X fl3 ^ a3 + 3 + 3 + 3 _ ^3x4 — a^2^ 

and, in general, from the definition of an exponent 
(«»»)" —. a^ X a^ X a"^ X a^ . . . =a^X^\ 



204 ELEMENTS OF ALGEBKA. [CHAP. VIL 

hence, the n*^ power of the m^^ power cf a number is equal to the 
mn^^ power of this number. 

Let us see if the converse of this is also true. 

Let V V^ - ^ ' 

then raising both members to the n^^ power, we have, from the 
definition of the n^^ root, 

"l^= b"" ; 
and by raising both members of the last equation to the m*^ power 

a — S"*". 
Extracting the mn^^ root of both members of the last equation, 

we have, *" V^ - ^ ' 



and hence, \/^J~a =i "^^Ta^ 

since each is equal to b. Therefore, the n^'^ root of the 7fi>*^ root 
of any number^ is equal to the mn*^ root of that number. And 
in a similar manner, it might be proved that 



By this method we find that 

1. t/256 = x/y25Gz=./l6 



2. 6^2985984 = y^^ 2985984 = l/vm = 12. 

3. yi771561 = ^y/ 1771561 = 11. 

4. y 1679616 = yi296z^v/y/n[296 = 6. 

Remark. — Although the successive roots may be extracted in 
any order whatever, it is better to extract the roots of the lowest 
degree first, for then the extraction of the roots of the higher 
degrees, which is a more complicated operation, is effected upon 
uumbers containing fewer figures than the proposed number. 



• 



CHAP. VII.] EXTRACTION OF ROOTS. 205 

Extraction of Boots hy Approximation. 

144, When it is required to extract the n*^ root of a number 
which is not o. perfect n^^ power ^ the method already explained, will 
give only the entire part of the root, or the root to within less 
than 1. As to the part which is to be added, in order to com 
plete the root, it cannot be obtained exactly, but we can approx- 
imate to it as near as we please. 

Let it be required to extract the n*^ root of a whole number, 

denoted by a, to within less than a fraction — ; that is, so near, 

*^t the error shall be less than — . 

P 
We observe, that we can write 

aw" 

a = — — . 

If we denote by r the root of the greatest perfect w** power in 

d X V^ ^^ 

ap^f the number — = a, will be comprehended between — and 

( y 4- 1)" /- 

- — ^ ; therefore, the ^a will be comprised between the 

r r -\- \ 

two numbers — and ; and consequently, their difference 

1 r 

— will be greater than the difference between — and the true 
P P 

r 
root. Hence, — is the required root to within less ':han the 

P 

fraction — : hence, . 
P 

To extract the n^^ root of a whole number to within less than 
a fraction — , multiphj the number by p^ ; extract the n*^ root of 
the product to within less than 1, and divide the result by p. . 

Extraction of the n^^ Root of Fractions. 

145« Since the n*^ power of a fraction is formed by raising 
both terms of the fraction to the n*^ power, we can evidently 
find the n*^ root of a fraction by extracting the n*^ root of 
both terms. 



206 ^ ELEMENTS OF ALGEBRA. [CHAP. VIL 

If both terms are not perfect ti*'^ powers, the exact n*^ root 
cannot be found, but we may find its approximate root Uf 
within less than the ffactio7ial unit, as follows: — 

Lot y represent the given fraction. If we multiply both 

terms by 

0" ^ it becomes, — = , 

' b b"" 

Let r denote the n^^ root of the greatest n*^ power in a6"~*', 

ah^~^ r» {r 4- IV 

then — - — will be comprised between — and ^ ' -' - 



T Cb 

and consequently, — will be the n^^ root of — - to within les» 



than the fraction — ; therefore. 

Multiply the numerator by the {n — 1)^^ power of the denorni 
nator and extract the n^^ root of the product: Divide this root 
by the denominator of the given fraction, and the quotient will 
he the approximate root. 

When a greater degree of exactness is required than that 
indicated by — , extract the n^^ root of a5«-i to withir ^.ny 

1 r' . r' 

fraction — ; and desimiate this root by — . Now, sinci — 
J9 ' ° p P 

is the root of the numerator to within less than — , it fob - -ws, 

p 

r' . . . 1 

that T— is the true root of the fraction to within less thar. -— 
hp bp 

EXAMPLES. 

1. Suppose it were required to extract the cube root d 15 

to within less than — . We have 

15 X 123 ^ 15 X 1728 = 25920. 
Now, the cube root of 25920, to within less than I is ^ 
hence, the required root is, 

12~ 12' 



CHAP. VII.] EXTRACTION OF ROOTS. 207 

2. Extract the cube root of 47, to within less than — -. 

We have, 

47 X 203 ^ 47 X 8000 = 376000. 



Now, the cube root of 37G000, to within less than I, is 72; 

r2 12 1 

2-0 = ^20' '^^^'^^^ ^"'^'^"^ 20- 



hence, ^47 r= — = 3 :r^, to within less than ^^ 



3. Find the value of y^ 25, to within less than .001. 

To do this, multiply 25 by the cube of 1000, or 1000000000, 
which gives 25000000000. Now, the cube root of this number. 
is 2920 ; hence, 

y^ — 2.920 to within less than .001. 
Hence, to extract the cube root of a whole number to 
within less than a given decimal fraction, we have the following 

RULE. 

Annex three times as many ciphers to the number^ as there are 
decidual places in the required root ; extract the cube root of the 
number thus formed to luithin less than 1, and point off from 
the right of this root the required number of decimal places. 

146. We will now explain the method of extracting the cube 
root of a decimal fraction. 

Suppose it is required to extract the cube root of 3.1415. 

Since the denominator, 10000, of this fraction, is not a per 
feet cube, make it one, by multiplying it by 100 ; this is equiva 
lent to annexing two ciphers to the proposed decimal^ which theu 
becomes, 3.141500. Extract the cube root of 3141500, that is, 
of the number considered independent of the decimal point to 
within less than 1 ; this gives 146. Then dividing by 100, oi 
^1000000, and we find, 

y 3.1415 — 1.46 to within less than 0.01. 
Hence, to extract the cube root of a decimal fraction, we havi 
the following 



208 ELEMENTS OF ALGEBRA. ICHAP. VIL 

RULE. 

A7inex ciphers till the whole number of decimal places is equal 
to three times the number of required decimal places in the root. 
Then extract the root as in whole numbers^ and point off the re- 
quired number of decimal places. 

To extract the cube root of a vulgar fraction to within less 
than a given decimal fraction, the most simple method is, 

To reduce the proposed fraction to a decimal fraction^ continuing 
the division until the number of decimal places is equal to three 
times the number required in the root. 

The question is then reduced to extracting the cube root of 
a decimal fraction. 

Suppose it is required to find the sixth root of 23, to 
within less than 0.01. 

Applying the rule of Art. 144 to this example, we multiply 
23 by (100)^, or annex twelve ciphers to 23 ; then extract the 
sixth root of the number thus formed to within less than 1, 
QXid divide this root by 100, or point off two decimal places 
ou the right : we thus find, 

^/23 =: 1.68, to within less than 0.01. 

EXAMPLES. 

1. Find the ?/473 to within less than gV ^^^- "^f* 

2. Find the ^/79 to within less than .0001. Ans. 4.2908. 

3. Find the ^/l3 to within less than .01. Ans. 1.53. 

4. Find the ^3.00415 to within less than .0001. 

Ans. 1.4429. 

5. Fmd the ^0.00101 to within less than .01. 

Ans. 0.10. 

6. Find the l/T¥ to within less than .001. Ans, 0.824. 



25 



CHAP. VII.j EXTRACTION OF ROOTS. 209 

Extraction of Roots of Algebraic Quantities, 

147« Let us first consider the case of monomials, and in cider 
to deduce a rule for extracting the n^^ root, let us examine the 
law for the formation of the n^^ power. 

From the definition of a power, it follows that each factor 
of the root will enter the power, as many times as there are 
units in the exponent of the power. That is, to form the n** 
power of a monomial, 

We form the n'* power of the co-efficient for a new co-efficient^ 
and write after this, each letter affected with an exponent equal to 
u times its primitive exponent. 

Conversely, we have for the extraction of the w'* root of a 
monomial, the following 

RULE. 

Extract the n'^ root of the numerical co-efficient for a new co- 
efficient, and after this write each letter affected with an exponent 

equal to — th of its exponent in the given monomial ; the result 
will be the required root. 



Thus, ^64a963c6 = ^a^bc^ ; and *^ 16a%^^c* = 2a^h, 

From this rule we perceive, that in order that a monomial 
may be a perfect n'* power: 

1st. Its co-efRcient must be a perfect n** power; and 

2d. The exponent of each letter must be divisible by n. 

It will be shown, hereafter, how the expression for the root 
of a quantity, which is not a perfect power, is reduced to its 
simplest form. 

148. Hitherto, in finding the power of a monomial, we have 
paid no attention to the sign with which the monomial may be 
affected. It has already been shown, that whatever be the sign 
of a monomial, its square is always positive. 

14 



210 ELEMENTS OF ALGEBRA. LCHAP. VIL 

Let n be any whole number; then every power of an even 
degree, as 27z, can be considered as the n*^ power of the square; 
that is, {a?Y = ^"^ • hence, it follows, 

That every power of an even degree^ will he essentially posi 
itve, whether the quantity itself be positive ^r negative. 

Thus, (±2a263c)4 ^ 4. iQa^^W 

Again, as every power of an uneven degree, 2?i + I, is but 
the product of the power of an even degree, 2?i, by the first 
power ; it follows that. 

Every povjer of a monomial^ of an uneven degree^ has the same 
sign as the monomial itself. 

Hence, (+ 4a25)3 = -\- 64a^P ; and ( — 4.a^Y — — Q4a^b\ 

From the preceding reasoning, we conclude, 

1st. That when the index of the root of a monomial is uneven^ 
iJie root will be affected with the same sign as the monomial. 

Thus, 

y + 8a3 = + 2a ; ^^ - Sa^ = — 2a ; ^ - S2a^%^ = — 2a^, 

2d. When the index of the root is even, and the monomial a 
positive quantity, the root has both the signs -f «^c? — . 

Thus, 1S^81a*5i2= ± Za¥ ; ^ 64ai8 ^ _+. ^a^. 

3d. When the index of the root is even, and the monomial nega- 
live, the root is impossible; 

For, there is no quantity which, being raised to a power of 
aa even degree, will give a negative result. Therefore, 

\/~=^' '/-^' '/-^' 
are symbols of operations which it is impossible to execute 
Tliey are imaginary expressions. 

EXAMPLES. 

1. What is the cube root of Sa^iV^I Ans. 2a%C^, 

2. What is the 4th root of Qla^b^c^^l Ans. Sab^c*. 

3. What is the 5th root of — S2a^c^od^^ 1 Ans. — 2ac'^d?. 
A. What is the cube root of — ^^tia^^c^l Ans. — baWc. 



CHAP. VII.] EXTRACTION OF ROOTS. 211 

Extraction of the n"* Root of Polynomials. 

148t Let N denote any polynomial whatever, arranged with 
reference to a certain letter. Now, the n^^ power of a poly- 
nomial is the continued product arising from taking the poly, 
nomial n times as a factor: hence, the first term of the pro- 
duct, wlit-n arranged with reference to a certain letter, is the 
n'* power of the first term of the polynomial, arranged with 
reference to the same letter. 

Therefore, the 7i'* root of the first term of such a product, 
will be the first term of the n*^ root of the product. 

Let. us denote the first term of the n^^ root of N by r, 
and the following terms, arranged with reference to the lead- 
ing letter of the polynomial, by r\ r", /", &c. We shall 
have, 

N= {r -\- r' -\- r" -\- . . &c.)» ; 

or, if we designate the sum of all the terms after the first 

DV 5, 

iV'= {r + s)" = r^ -\- nr^-'^s + &c., 

_ ^n _|_ ^y.n-1^^' _j_ ^'f _|. ^Q^ -J _|_ ^Q 

If now, we subtract r" from iV, and designate the remainder 
by i?, we shall have, 

R = N —r"" = nr^'-h'' + m-"- V + &;c., 

w^hich remainder will evidently be arranged with reference to 
the leaching letter of the polynomial ; therefore, the first term 
will contain a higher power of that letter than either of the 
succeed hig terms, and cannot be reduced with any of them. 
Hence, if we divide the first term of the first remainder, by 
n times the (n — 1)'* power of the first term of the root, the 
quotient will be the second term of the root. 

If now, we place r -\- r' = u, and denote the sum of the sue. 
feeding terms of the root by 5', we shall have, 



212 ELEMENTS OF ALGEBRA [CHAP. VIL 

If DOW, we subtract u^ from iV", and den;te the remainder by 
E\ we shall have, 

i2' = iV— «• = n(r -f- ?•')«- V + &c., ' 

= ?2r«-i(r" + r'" + &c. ) + &c., 

-Ii we divide the first term of this remainder by n times 
the (n — 1)*^ power of the first term of the root, we shall 
have the third term of the root. If we continue the operation, 
we shall find that the first term of any new remainder, divided 
by n times the (n — 1)'* power of the first term of the root, 
will gi\ e a new ter.m of the root. 

It nitty be remarked, that since the first term of the first 
remain dsr is the same as the second term of the given poly- 
noriiaJ, we can find the second term of the root, by dividing 
the seoond term of the given polynomial by n times the 
(;i — 1)'* power of the first term. 

ITen^e, for the extraction of the n'* root of a polynomial, 
we have the following 

RULE. 

I. Arrange the given polynomial with reference to one of its letters, 
and extract the n'* root of the first term; this will be the first 
term of the root. 

II. Divide the second term by n times the {n — 1)'* power of the 
first term of the root ; the quotient wiube the second term, of the root 

III. Subtract the n'* power of the sum of the two terms already 
found from the given polynomial^ and divide the first term of 
the remainder by n times the {n — 1)'* power of the first term of 
the root ; the quotient will be the third term of the root. 

IV. Continue this operation till a remainder is found equal to 
0, Of, till one is Jound whose first term is not divisible by n times 
the (k — 1)'* power of the first term of the root: in the former case 
the root is exact, and the given polynomial a perfect n^^ power ; 
hi the latter case, the polynomial is an imperfect n*^ power. 



CHAP. VII.] EXTRACTION OP ROOTS. 218 

149. Let us apply the foregoing rule to the following 

EXAMPLES. 

1. Extract the cube root of a:6—6:c5+ 15a;*— 20^3-1- 15aj2—6.rfl. 

(x^-2xy=x^-Qx^-^12x*— Sx^ Sx* 

1st rem. 3:c*-12a:3+ &c. 

(^x^-2x-{-iy=x^^Qx^-^lbx'^-20x^-\-lbx^—Qx-{-l. 

In this example, we first extract the cube root of x^, which 
gives x^, for the first term of the root. Squaring x^, and mul- 
tiplying by 3, we obtain the divisor Sx* : this is contained in 
the second term — Qx^, -A2^ times. Then cubing the part of 
the root found, and subtracting, we find that the first term of 
the remainder 3a;*, contains the divisor once. Cubing , the v/hole 
root fotmd, we find the cube equal to the given polynomial. 
Hence, x^ — 2x-\- 1, is the exact cube root. 

2. Find the cube root of 

a;6 4_ Qx^ _ 40a:3 -f. 96a; — 64. 

3. Find the cube root of 

8x^ - 12a;5 + 30a;* - 25a:3 4. 30^52 __ i2a; + 8. 

4. Find the 4th root of 16a*— 9Qa^x + 21Qa^x'^ — 216aa;S -f 81a;* 

16a*-96a3a;+216a2a;2_216aa;3 + 81a;4 2a-3a; 
(2a-3a;)*= lQa^-9Qa\x+21Qa^x''-216ax^+Slx* k X (2a)a=32«3. 

We first extract the 4th j-oot of 16a*, which is 2a. We then 
raise 2a to the third power, and multiply by 4, the index of the 
root ; this gives the divisor 32a3. This divisor is contained in 
the second term — dQa'^x, — 3a; times, which is the second term 
of the root. Raising the whole root found to the 4th power 
we find the power equal to the given polynomial. 

5. What is the 4th root of the polynomial, 

81 aV + 166*6?* - 9QahPd^ •- 21Qa^c^d + 2lQa^c^b^d^. 

6. Find the 5th root of 

32a;5 — 80a:4 ^ 80a;« - 40a;2 + 10a; - 1. 



I 

214 ELEMENTS OF ALGEBRA. LCHAP. VIL 

Transformation of Radicals of any Degree. 

150. The principles demonstrated in Art. 104, are general. 
For, let ^.Ja^ and 'i/^ be any two' radicals of the ?i'* degree, 
and denote their product by p. We shall have, 
V^xy^=i9 - - - (1). 
By raising both members of this equation to the «.** power, 
we find 

(Xf^Y X {\n>Y =i3", or ah =^5"; 

whence, by extracting the n^^ root of both members, 

%fab=P - - - (2). 
Since the second members of equations (1) and (2) are the 
same, their first members are equal, whence, 

'Lfa X \rb — ''J'ab : hence, 

1st. The product of the n^^ roots of two quantities^ is equal to 
the n*^ root of the product of the quantities. 

Denote the quotient of the given radicals by q^ we shall have 
^=q .... (1); 
and by raising both members to the n^^ power, 

(\A)" • 

whence, by extracting the n^^ root of the two members, we 
have, f 

'~a 



j = ? - - - - (2). 

The second members of equations (1) and (2) being the same, 
iheir . first members are equal, giving 



— hence, 

2d. The quotient of the n^^ roots of two quantities^ is equal to 
tliC n** root of the quotient of the quantities. 



i 



CHAP. VII.l TRAXSFOEMATION OF RADICALS. 215 

151, Let us apply the first principle of article 150, to the 
simplification of the radicals in the following 

EXAMPLES. 



1. Take the radical ^/54a*6^. This may be written, 

2. In like manner, 

^/8^=i 2 y^ ; and y^iSo^W == 2ah\ \f^^? ; 

3. Also, ^ 

^192a^^ci2 ^ ^64a6ci2 x ^806 = 2ac2 6^3T6'. 

In the expressions, Zah \J 2ac^, 2 •l/o^, 2a62c */ Sctc^, 

each quantity placed before the radical, is called a co- efficient 
of the radical. 

Since we may simplify any radical in a similar manner, we 
have, for the simplification of a radical of the w** degree, the 
following 

RULE. 

Resolve the quantity under the radical sign into tiuo factors^ one 
of ivhich shg.ll he the greatest perfect n*^ power which enters it; 
extract the n*^ root of this factor, and write the root without the 
radical sign, under 2chich, leave the other factor. 

Conversely, a co-efficient may be introduced under- the radical 
sinn, .by simply raising it to the n*^ power, and writing it as a 
factor under the radical sign. 

Thus, ^ah \J~^M^ = 3^/27^^ X ^2^ = ^54a*6V. 

152. By the aid of the principles demonstrated in article 143, 
we are enabled to make another kind of simplification. 

Take, for example, the radical %J~\o^\ from the principles re* 
ferred to, we have, 



\f^^ ^ Y^/4^, 



216 ELEMENTS OF A.LGEBRA. [CHAP. VII. 

and' as the quantity under the radical sign of the second degree 
is a perfect square, its root can be extracted : hence, 

Ill like manner, 
In general, 



that is, when the index of a radical is a multiple of any number 
n, and the quantity under the radical sign is an exact n** power. 
We con^ without changing the value of the radical^ divide its index 
by n, and extract the n^^ root of the quantity under the sign, 

153. Conversely, The index of a radical may he multiplied by 
any number^ provided we raise the qvantity under the sign to a 
power of which this number is the exponent. 

For, since a is the same thing as 'i/a", we have, 



"^fa— xl ^To^ — "^'iJ'aK 



154» The last principles enable us to reduce two or more 
radicals of different degrees, to equivalent radicals having a com- 
mon index. 

For example, let it be required to reduce the two radicals 

3^/2^ and \f{(^-\-h) 
to the same index. 

By multiplying the index of the first by 4, the index of the 
second, and raising the quantity 2a to the fourth power; then 
multiplying the index of the second by 3, the index of the 
first, and cubing a ■\- b^ the value of neither radical vill bf* 
changed, and the expressions will become 

\J~^ = 12^2%^ = ly^re^; and y (a + 6) = '^ (a + b)\ 

and similarly for other radicals: hence, to reduce radicals to a 
common index, we have the following 



CHAP. VII.] TKANSFORMATION OF RADICALS. 217 

HULK 

Multiply the index of each radical hy the product of the indices 
of all the other radicals^ and raise the quantity under each radical 
sign to a power denoted by this product. 

This rule, which is analogous to that given for the reduction 
of fractions to a common denominator, is susceptible of similar 
iDodifications. 

For example, reduce the radicals 

to a common index. 

Since 24 is the least common multiple of the indices, 4, 6, and 
8, it is only necessary to multiply the first by 6, the second by 
4, and the third by 3, and to raise the quantities under each rad 
ical sign to the 6th, 4th, and Sd powers, respectively, which gives 

Addition and Subtraction of Radicals of any Degree. 

155, We first reduce the radicals to their simplest form by 
the aid of the preceding rules, and then if they are similar^ in 
order to add them together, we add their co-efficients^ and after 
this sum write the common radical; if they are not similar, th*^ 
addition can only be indicated. 

Thus, 3^+ 2^6"== 5^. 

\ EXAMPLES. 

1. Find the sum of ^/iS^ and b ^/Iba, Ans. 95y^a. 

2. Find the sum of 3 ^^^ and 2 ^/^^. Ans. 5 ^/2a. 

3. Find the sum of 2 y^ and 3 ^5". ' Ans. 9 y^. 

155*. In order to subtract one radical from another when 
they are similar, 

Subtract the co-efficient of the subtrahend from the co-efficient of 
(he minuend, and write this difference before the common radical^ 



I 



218 ELEMENTS OF ALGEBRA. .CHAP. YU. 

Thus, 8a y^ - 2c \fh == (Sa - 2c) ^T; 

but, 2a6 y'^cZ — 5a6 -/^ are irreducible. 

1. From y8a36 + 16a* subtract ^ 6* + 2a63. 



^715. (2a - I) y6 + 2a. 
2. From 3 6^402" subtract 2\f2a. Ans. \f2a. 

Multiplication of Radicals of any Degree. 

156. We have shown that all radicals may be reduced to 
equivalent ones having a common index; we therefore suppose 
this transformation made. 

Now, let a \/~b and c '^.J~d denote any two radicals of the 
same degree. Their product may be denoted thus, 

a y^ X c "yZ; 
or since the order of the factors may be changed without affect- 
ing the value of the product, we may write it, 

ac X "y^X \fd or (Art. 150), since '^'Zx y^= \/^; 
we have finally, 

a'^JT ^ c')J~d — ac\fhd \ 
hence, for the multiplication of radicals of any degree, we have 
the following 

RULE. 

I. Reduce the radicals to equivalent ones having a common index. 

l\' Multiijly the co-efficients together for a new co-efficient; after 
this write the radical sign with the common index, placing under 
it the product of the quantities under the radical signs in the two 
factors; the result is the product required. 

EXAMPLES. 

1. The product 



;/z±? X - 3. ;^^i2!+i!)! = _ 6a^ y@ 



cd 

_ 6a2 (a2 _{_ 52) 



CHAP. VII. J TRANSFORMATION OF RADICALS. 219 

2. The product 

3. The product 

-^-yy X — y y-— y — . 

4. The product 

5. Multiply /2x 3/3 by t/l x ^^. 

6. Multiply 2y^ by Z\f^. 

Ans. 6 6/337500. 

^ /2' /3~ 

7. Multiply 4y y by 2y— . 

10 /27" 
^"^' ^ V 256- 

8. Multiply ^, ^/3", and y^, together. 

Ans. 1^648000. 

9. Multiply \/-^, \/i7 and i*/6, together. 

'2 



^"'- ■■27- 



42 fi 

10. Multiply (4^y^+5\/l) by (y/X + 2^1). 



^.,| + l?y-42. 

Division of Radicals of any Degree. 

157» We will suppose, as in the last article, that the radicals 
have been reduced to equivalent ones having a common index. 

Let d'Lfb^ and c!!/^ represent any two radicals of the 
n • degree. Tlie quotient of the first by the second may be 
written, 

a iff a \fh 

■ V — >^ _v 



220 



ELEMENTS OF ALGEBRA. 



[CHAP. VIL 



or, since 






(Art. 150), «re have, 



c \fd c \l d 
Hence, to divide one radical by another, we have the fbl 
lowing 

RULE. 

I. Reduce the radicals to equivalent ones having a common index, 

II. Divide the co-efficient of the dividend hy that of the divi- 
sor for a new co- efficient ; after this write the radical sign with 
the common index, and place under it the quotient obtained hy 
dividing the quantity under the radical sign in the dividend by that 
in the divisor ; the result will be the quotient reqvAred. 

EXAMPLES. 



a2_62 



1. What is the quotient of c yaW -f b^ divided by d 
c^^ l/aW -\-b^ __ ^3 /86 (aW -f b^) _ 2cb^ I a^ + b'^ 



d 3 /^2 _ 52 ~ dy a 



2 — 62 



W 



2. Divide 2y^xyT by 1^/2 X^^. 



Ans. 412/288. 



3. Divide y^i X 2 3/3 by y/^\/ix^. 



^"^- VVi 



^Divide 1/1 by (72 + Sy^). 



Ans. 



10* 



5. Divide 1 by \/^+\/T. 

Ans, \A^-V^+\/^-V^ 
a — b 



CRAP Vir.l TRANSFORMATION OF RADICALS 221 

6. Divide \/~^' + \fb by ^^ _ ^ 

Ans. — -^ — 

a — 

Formation of Powers of Radicals of any Degree. 

158. Let a "ifb represent any radical of the n^^ degree, 
rhen we may raise this radical to the m*^ power, by taking 
it m times as a factor; thus, 

a yTx a y^ a y^ 

But, by the rule for multiplication, this continued product is 
eqMal to a"* \fb^\ whence, 

(a y^)*" == aJ^ \fh^ .... (1). 
We have then, to raise a radical to any power, the following 

RULE. 

Raise the co-efficient to the required power for a new co-efficient ; 
after this write the radical sign with its primitive index, placing 
under it the required power of the quantity under the radical 
sign in the given expression ; the result will be the power required. 

EXAMPLES. 

1. (y4a3)2 = %/JI^Y = j^/T6^ = 1a\J~^- 2a -^/a. 

2. (33^/2^)5 = 35 y^)^ =12433^/32^= 486a y^^. 

When the index of the radical is a multiple of the expo- 
nent of the power to which it is to be raised, the result can 
be simplified. 

For, */ 2a = v/v/^^ (Art. 152): hence, in order to square 

*/ 2a, we have only to omit the first radical sign, which gives 

Again, to square l/36, we have l/36 = \.'l/i^b'. hence^ 
(y^)2 = ^/36; hence, 



J 



222 ELEMENTS OP ALGEBKA, [CHAP. VIL 

When the index of^ the radical is divisible by the exponent of 
the power to which it is to be raised, perform the division, leaving 
the quantity under the radical sign unchanged. 

Extraction of Roots of Radicals of any Degree. 

J 59. By extracting the m^^ root of both members of equa- 
tion (1), of the preceding article, we find, 



Whence we see, that to extract any root of a radical of any 
degree, we have the following 



RULE. 



Extract the required root of the co-efficient for a new co-efficient; 
after this write the radical sign with its j^rmziive index, under 
which place the required root of the quantity under the radical 
sign in the given expression ; the result will be the root required. 



EXAMPLES. 



1. rind the cube root of 8^^. Ans. 2y^. 

2. Find the fourth root of —3/256. Ans. -^V^- 



16V 2 

159*. If, however, the required root of the quantity under the 
radical sign cannot be exactly found, we may proceed in the 
following manner. If it be required to find the m** root of 
cV^ the operation may be indicated thus. 



y^c «y/7 = y7 y^'i/d', 



but \/ !i/j = ""l/^j whence, by substituting in the previousi 
equation, 

Consequently, when we cannot extract the required root of the 
quantity under the radical sign, 



%^ 



CHAP. VII.] TRANSFORMATION OF RADICALS. 223 

Extract the required root of the co-efficient for a new co-efficient i 
after this^ write the radical sign^ with an index equal to the in o- 
duct of its primitive index by the index of the required rooty 
leaving the quantity under the radical sign unchanged, 

EXAMPLES. 

When the quantity under the radical is a perfect power, of 
the degree of either of the roots to be extracted, the result can be 
simplified. 

Thus, 1/ 1/8^ = y^ -^/S^ = \f2a. 

In like manner, kI ^J *^o? — sj ,J~W — iT^a. 

2. Find the cube root of ^-y/^- ^w^. — -/3. 



1 /7^^ ' . 1 



3. Find the cube root of —^2^. Ans, — ^/2aP. 

Different Roots of the same Power. 

160. The rules just demonstrated depend upon the principle, 
that if two quantities are equal, the like roots of those quantities 
are also equal. 

This principle is true so long as we regard the term root 
in its general sense, but when the term is used in a restricted 
souse, it requires some modification. This modification io parti- 
cularly necessary in operating upon imaginary expressions, which 
are not roots, strictly speaking, but mere indications of opera- 
tions w^hich it is impossible to perform. Before pointing out 
these modifications, it will be shown, that every quantity has 
more than one cube root, fourth root, &c. 

It has already been shown, that every i[uantity has two square 
roots, equal, with contrary signs. 



224 ELEMENTS OF ALGEBRA. [CHAP. VIL 

1. Let X denote the general expression for the cube root of 

a, and let j9 denote the numerical value of this root; we have 

the equations 

x^ = a, and x^ = p^. 

The last equation is satisfied by making xz=zp. 
Observing that the equation X'^=zj)^ can be put under the form 
g.3 _ 2)3 —. Q^ and that the expression x^ — p^ is divisible by 
X — p, giving the quotient, x"^ -^ px ^ p"^^ the above equation can 
be placed under the form 

{x —p) {x^ -{-px -\-p^) = 0. 
Now, every value of x that will satisfy this equation, will 
satisfy the first equation. But this equation can be satisfied by 
supposing 

x — p = 0^ whence, x =p', 
or by supposing 

x^ + 2>^ + p^' = 0, 
from which we have. 



|.|yzr, „ - ->-'-^^ 



3, or x=pt ^ j; 



hence, we see, that thei-e are three different algebraic expressions 
for the cube root of a, viz : 

2. Again, solve the equation 

x*z=p\ 

m which p denotes the arithmetical value of */o. 

This equation can be put under the form 
2:4 _ ^4 _ . 
which reduces to 

{x^-p^){x^+p^) = 0', 
and tliis equation can be satisfied, by supposing 
2-2 _ ^2 _ Q . whence, x = dz p ; 
or by supposing 

3.2 _|_ ^2 _ 0^ whence, x =z ± ^ —p"^ = db p^ — 1. 



CHAr. VII.] TRANSFORMATION OF RADICALS. 225 

We therefore obtain four different algebraic expressions for the 
fourth root of a. 

3. As another example, solve the equation 
a;6 _ ^6 _ 0. 

This equation can be put under the form 
(a;3__j33)(^3_|.^3) _0; 
which may be satisfied by making either of the factors equal 
to zero. 

But, x^ — p^ = 0, gives 



.=^, and .=,(^1±V^). 

And if in the ev^uation x^ -\- p^ = 0, wb make ^ = — ^', it 
becomes x"^ — p'^ t= 0, from whicL we deduce 



x=zp\ and xz=p'\^ ^ j ; 

or, substituting for p' its value — p, 

. = -„ and . = -,( -'^/^ ). 

Therefore, x in the equation 

x^ — p^ = Oy 
and consequently, the 6th root of a, admits of six different alfft 
hraic expressions. If we make 

a = ^ , and a' = ^ , 

these expressions become 



P, «Pv ap, —p,— apy—a'p. 

It may be demonstrated, generally, that there are as many 
different expressions for the »** root of a quantity as there are 
units in n. If n is an even number, and the quantity is posi- 
tive, two of the expressions will be real, and equal, with con- 
trary eigns ; all the rest will be irjij^inary : if the quantity is 
negative, they will all be 'maginary. 

15 



226 ELEMENTS OF ALGEBRA. [CHAP. VIL 

If n is odd, one of the expressions will be real, and all the 
rest will be imaginary. 

161. If in the preceding article we niake a = 1, we shall find 
the expressions for the second, thirl, fourth, &c., roots of 1. 
Thus, 4- 1 and — 1 are the square" roots of 1. 

-1 --/^^^ 



Also, -f 1, -^- , and 

/it 

are the cv}>e roots of 1 : 



And +1, 1, + V ~ 1 ^^^ — -v/ — 1, are the fourth. 
roots of 1, &;c., &c. 

Rules for Imaginary Expressions. 

162. We shall now explain the modification of the rules for 
operating upon radicals when applied to irhaginary expressions. 



The product of V — a by y^ — a, by the rule of Art. 156, 
would be ^ -f- o?' Now, ^ -\- o? is equal to dz a, whence there 
is an apparent uncertainty as to the sign of a. The true pro-- 
duct, however, is — a, since, from the definition of the square 
root of a quantity, we have only to omit the radical sign, to 
obtain the quantity. 

Again, let it be required to form the product 

By the rule of Art. 156, we shall have 

but the true result is —J~ab, so lon^ as both the radicals 
J —a and .V — 6 are afl?*ected with the sign -f. 

For, yf^^a = ^, y ^^^T; and ^ — h = y^.^/ — 1 ; 
hence. 



CHAP. VII.J TRANSFORMATION OF RADICALS. 227 

In a similar manner, we treat all other imaginary expressions 
of the second degree ; that is, we first reduce them to the form 
of ay/ — 1, in which the co-efficient of ^/ —\ is real, and then 
proceed as indicated in the last article. 

162*. For convenience, in the application of the preceding 
principle, we deduce the different powers of -y/ — 1, as follows; 






(v-i)*={a/-i)^x(V-i)^= + i. 

The fifth power is evidently the same as the first power ; the 
sixth power the same as the second; the seventh the same as 
the third, and so on, indefinitely. 



163. If it is required to find the product of lV — a and 
*/ — 6, we should get, by applying the rule of Art. 156. 
1/ — a X %J~~ ^^ — 1/ + «6, hut this is not the true result. 
For, placing the quantities under the form 

\AxV^^=n: and 4^/6x1/^=^", 
and proceeding to form the product, we find 

\/"=^ X \A=T=: \/^ X \fb~x {\r^Y = \/~^X^/~-i, 

|, since, {i/ — ly = i\J y/ — \\ z=zy/ — 1 from the definition of 
a root. 

Hence, generally, when we have to apply the rules for radi- 
cals to imaginary expressions of the fourth degree, transform 
them, so that the only factor under the radical sign shall be 
— • 1, and then proceed as in the above example. 

1 ^-^^^^ 



Let us illustrate this remark, by showing mat 

is an expression for the cube root of 1, or that, in the restricved 
feense^ it is a cube root of 1. 



1 



228 ELEMENTS OF ALGEBRA. I CHAP. VIT 

We have 



2 

8 
-- 1 4- 3 -y/S". ^/^^ -3x-3-3-v/3. i/=T _ _8^ _ , 

8 ~ 8 - ■^• 

_ 1 _ ^ _ 3 

In like manner, we may show, that ^ is another 

expression for the cube root of 1, when understood in the 
restricted sense. It may be remarked that either of these ex- 
pressions is equal to the square of the other, as may easily 
be shown. 

Of Fractional and Negative Exponents, 

164, We have yet to explain a system of notation by means 
of which operations upon radical quantities may be greatly 
amplified. 

We have seen, in order to extract the n^^ root of the quan- 
tity a*", that when ?/t is a multiple of n, we have simply to 
divide the exponent of the power, by the index of the root to 
be extracted, thus, 

\J a^ = a". 

When m is not a multiple of n^ it has been agreed to 
retain the notation, 



these two being regarded as equivalent expressions, and both 
indicating the n*^ root of the m^^ power of a, or what is the 
same thing, the m*^ power of the n*^ root of a ; and generally, 

When any quantity is written with a fractional ex])onent, the 
numerator of the fraction denotes the poiver to which the quantity 
%a to he raised, and the denominator indicates the root of this 
power which is to he extracted. 



i 

CHAP. VIL] THEORY OF EXPONENTS. 229 

165i We have also seen that a"» may be di voided by a", 

when m and n are whole nunr^bers, by simply subtracting n 

from w, giving 

a"* '^ 

— = a"^" = aP ; 

a" ' 

in which we have designated the excess of m over n by p. 

Now, if n exceeds m, p becomes negative, and the exaiot 

division is impossible ; but it has been agreed to retain the 

notation 

a" 
But when m<^n^ in the fraction, 

. ^' 
we may divide both terms by a*", and we have 
«"» 1 1 



hence, a-^ is equivalent to — , and both denote the recipro. 

cal of aP. 

We have, then, from these principles, the following eqiii?». 
lent expressions, viz. : 



\fa equivalent to a" 



\f^ or («y^)'" 


a 


a». 


1 

a" 


a 


ar^. 


1 n/r 

or K — 
ya V a 


u 


_ 1 
a « 


1 « /r 


u 


m 

a « 


166. It has been shown above that 


1 _ 



- = a—" ; if now we 
divide 1 by both members of this equation, we shall have, 
«"=—-: hence, we conclude that, 



230 ELEMENTS OF ALGEBRA. [CHAP. VII 

Any factor may he transferred from the numerator to the de- 
nominator, or from the denominator to the numerator, by changing 
ike siyn of its exponent. 

167. It may easily be shown that the rules for operating 
lipon quantities when the exponents are positive whole numbers, 
aie equally applicable when they are fractional or negative. 

In the first place, it is plain that both numerator and 
denominator of the fractional exponent^ may be multiplied by 
the same quantity without altering the value of the expression, 
since by definition the m** power of the m*^ root of a quan- 
tity is equal to the quantity itself. This principle enables us 
to reduce quantities, having fractional exponents, to equivalent 
ones having a common denominator. 

m r 

Let it be required to find the product of a" and a** 

m r ms nr 

We have, a"" x a^ = a"* X a"*' 

m (Art. 164), ns/ ^ms y^ nsy^ .= «y^"«s 4- nr * 

ms + nr 

This last result is equivalent to a "* ' hence, 

m r ms-\-vr ^ 

a"^ X a^ = a ^^ ' 

the same result that would have been obtained by the appli- 
cation of the rule for the multiplication of monomials, when 
the exponents are positive whole numbers. 
If both exponents are negative, we shall have, 

_^ -L 11 1 ms + nr 

a "Xc. * = — X — = X- - = ^^ "» * 

jrt r VIS -f- vr 

a" a* a ^^ 
If one of the exponents is positive, and the other negati-^e, 
tr« shall have, 

m r tTi 1 VIS 1 

a " X a" *" = a~ X — = a""* X -^ , 
a' a** 



w ■ 

CHAP Vn.] THEORY OF EXPONENTS. 231 



ns / \ 



ns / n^is 



whence, «v/ a'"^ X y -^f— V —^r;: = "V<^"'* ""'' = « "* ' 



7ns — -ar 



and finally, a« x a « = a "* 

We have, therefore, for the multiplication of quantities when 
the exponents are negative or fractional, the same rule as when 
they are positive whole numbers, and consequently, the same 
rule for the formation of powers. 

EXAMPLES. 

\ . ■ . 

^ -1 23 lJi_2 

1. a^6 2^1 y.a^h'c^ =a^ h^c \ 

s 2. Sa-^'^ X 2a"^6 V = QaT ^ b^c^ 

_i i _L 

1 

4. Find the square of fa^. 

We have, (fa*)' = (if X a*^ ' = i J. 

1 5 

5. Find the cube of ^a". Ans. ^ar, 

m r 

168. Let it be required to divide a" by a» . We shall have, 

m m 

~ m r 'Z ms — rn 

a 



— =:a" X a * or (Art. 167), — = « "* 
a^ a* 

If both exponents are negative. 



= a "^ X a* = a "* ' by the last article. 

If one exponent is negative, 

~ TO jr ms -f- rn 

=a" X a''=: a »* ' by the preceding article. 



232 ELEMENTS OF ALGEBRA. [CHAP. VII. 

Hence, we see that the rule for the division of quantities, 
'with fractional exponents, is the same as though the exponents 
were positive whole numbers; and consequently we have the 
same rule for the extraction of roots, as when the exponents are 
positive whole numbers. 

EXAMPLES. 

2. 

3. a^ X b^~ar^f^=a^b~^. 



i_ 

20 



^ •§- -X I JU -A 



4. Divide ^2(}¥c^ by ^aH^c 2\ " Ans. 4a^6c*, 

5. Divide Q^a^b^c'^ by Z2a-%~^c~~^. Ans. 2a^^b\ 

3 /~2: 1 

a^ = a^ ; 7. 

_3 3 r^^T 1 _2. 

8. Ua^=a ^'^ 9. UaH-'' = aH ^ 

169. We see from the preceding discussion, that operations to 
be performed upon radicals, require no other rules than those 
previously established for quantities in which the exponents are 
entire. These operations are, therefore, reduced to simple oper 
ations upon fractions, with which we are already familiar. 

GENERAL EXAMPLES. 

1. Keduce — ^ ■ to its simplest terms. 

Ans. 4.1/W, 

2. Reduce ■< j- > to its simplest terms. 

( 2l/2(3f ) 



Ans. -L 3^3. 



CHAP VII. J THEOEY OF EXPONENTS. 233 



3. Keduce / I — —-^ V to its simplest terms. 

2/2. (#i 




4. What is the product of 

5 13^2 lis. 1.x 

a^ 4- a^^ + aH^ -^ ab + aH^ + b\ by J - 6^ 

Ans. a^ — b^. 

5. Divide a^ — a^"^ - Jb + 6^, by J - b~^. 

Ans. d? — 6. 

170. If we have an exponent which is a decimal fraction, as. 
for example, in the expression 10 ' ^^^ from what has gone bf* 

301 

fore the quantity is equal to (10)^^°°' or to ioo^/^0)^^S the 
value of which it would be impossible to compute, by any process 
yet given, but which will hereafter be shown to be nearly equal 
to 2. In like manner, if the exponent is a radical, as JW^ VT^» 
&c., we may treat the expression as though the exponents were 
fi-actional, since its values may be determined, to any degree of 
exactness, in decimal terms. 



CHAPTER VIL: 

OF SER1E<5^ — ARITHMETICAL PROGRESSION GEOMETRICAL PROPORTION AND 

PROGRESSION RECURRING SERIES — BINOMIAL FORMULA SUMMATION OF 

SERIES PILING SHOT AND SHELLS. 

171 • A SERIES, in algebra, consists of an infinite number of 
terms following one another, each of which is derived from 
one or more of the preceding ones by a fixed law. This law 
is called the law of the series. 

Arithmetical Progression. 

n2. An ARITHMETICAL PROGRESSION is a scries, in which each 
term is derived from the preceding one by the addition of a 
constant quantity called the common difereiice. 

If the common difference is positive, each term will be greater 
than the preceding one, and the progression is said to be in 
creasing. 

If the common difference is negative, each term will be less 
than the preceding one, and the progression is said to be 
decreasing. 

Thus, ... 1, 3, 5, 7, . . . &c., is an increasing arithmetical 
'progression, in which the common \difference is 2 ; 

and 19, 16, 13, 10, 7, ... is a decreasing arithmetical 

progressio7i, in which the common difference is — 3. 

173. When a certain number of terms of an arithmetical 
progression are considered, the first of these is called the Jirst 
term of the progression, the last is called the last term «/ the 
•progression, and both together are called the extremes. All the 
terms between the extremes are called arithmetical means. Ail 
arithmetical progressii^n is often ^jailed a progression hg differences. 



CHAP. VIII.] ARITHMETICAL PROGRESSION. 235 

174i Let d represent the common difference of the arithmeti- 
cal progression, 

a.b.c.e./.ff.h.kf &c., 
which is written by placing a period between each two of the 
terms. 

From the definition of a progression, it follows that, 
b = a-{- d, c = b -{- d =z a -{- 2d, e = c-\-a = a-^Sd; 
and, in general, any term of the series, is equal to the first 
term plus as many times the -common difference as there are pre- 
ceding terms. 

Thus, Idt / be any term, and n the number which marks the 
place of it. Then, the number of preceding terms will be de- 
noted by n — 1, and the expression for this general term, will be 
I z=L a -\- {ii — \) d. 

If c? is positive, the progression will be increasing ; hence. 

In an increasing arithmetical progression, any term is equal to 
the first term, plus the product of the common difference by the 
number of preceding terms. 

If we make n z=z\, we have Z = a ; that is, there will be 
but one term. 

If we make 

71 = 2, we have I =La -\- d\ 
that is, there will be two terms, and the second term is equal 
to the first plus the common difference. 

EXAMPLES: 

1. If a = 3 and c? = 2, what is the 3d term? Ans, 7. 

2. If a = 5 and c? = 4, what is the 6th term *? Ans. 25. 

3. If a = 7 and d—b. what is the 9th term ? Ans. 47. 

The formula, 

l = a-\-{n-\)d, 
serves to find any term whatever, without determining those 
which precede it. , 



236 ELEMENTS OF ALGEBRA. [CHAP. VIIL 

Tims to find the 50th term of the progresskn, 
1 . 4 . 7 . 10 . 13 . 16 . 19, . . 

we have, ^ = 1 + 49 X 3 := 148. 

And for the 60th term of the progression, 

1 . 5 . 9. 13 . 17 . 21 . 25, . . . 

we have, ^ = 1 + 59 x 4 z= 237. 

174'*. If d is negative, the progression is decreasing, and the 
formula becomes 

l^z a — {n — 1) ^ ; that is. 

Any term of a decreasing arithmetical progression, is equal to 
the first term plus the product of the common difference by the 
number of preceding terms. 

EXAMPLES. 

1. The first term of a decreasing progression is 60, and the 
common difference — 3 : what is the 20th term % 

l = a-{n-l)d gives Z = 60 - (20 - 1) 3 = 60 - 57 == 3. 

2. The first term is 90, the common difference — 4 : what 
IS the 15th term? Ans. 34. 

3. The first term is 100, and the common difference — 2 • 
what is the 40th term? Ans. 22. 

175. If we take an arithmetical progression, 
a . b . c i . k . I, 

having n terms, and the common difference d, and designate 
the term which has p terms before it, by t, we shall have 

i = a-{-pd - - - - - (1). 
If we revert the order of terms of the progression, con. 
sidering I as the first term, we shall have a new progression 
whose common difference is —- d. The term of this new pro- 
gression which has p terms before it, will evidently be the same 
as that which has p terms after it in the given progression, 
and if we represent that term by t', we shall have, 

t'=zl—pd (2). 



CHAP. VIII.] ARirHMETICAL PROGRESSION. 237 

Adding equations (1) and (2), member to member, we find 
i-l- t' = a -{- 1 ; hence, 

The sum of any two terms, at equal distances from the extremes 
of an arithmetical progression, is equal to the sum of the extremes. 

176. If the sum of the terms of a progression be repre- 
sented by aS', and a new progression be formed, by reversing 
the order of the terms, we shall have 

Sz=a-{-b + c+ . . . . -\-i-{-k-\-l, 
S=l -i-k + i-^ . . . . +c-\- b + a. 
Adding these equations, member to member, we get 
2S = (a -hl)+ {b -{- k)+ (c -h i) . . . -\-{i + c)-{-{k-\-b)-\-{l-\-a); 

and) since all the sums, a -]- I, b + k, c -\- i , . . . are equal 
to each other, and their number equal to w, the number of 
terms in the progression, we have 

2S = (a -\- I) n, or S =1 — - — j n ; that is, 

The sum of the terms of an arithmetical progression is equal to 
half the sum of the two extremes multiiolied by the number of terms. 

EXAMPLES. 

1. The extremes are 2 and 16, and the number of terms 8: 
i^hat is the sum of the series '? 

S=l^-^\xn, gives AS'rr^^ti^X 8 = 72. 



/a-f l\ 



2. The extremes are 3 and 27, and the number of terms 12: 
what is the sum of the series 1 Ans. 180. 

3. The extremes are 4 and 20, and the number of terms 10: 
what is the. sum of the series'? Ans. 120. 

4. The extremes arc 8 and 80, and the number of terms 10: 
what is the sum of the series 1 Ans. 440. 

The formulas 

l = a + {n-\)d and S=(^-^^y:n, 



238 



ELEMENTS OF ALGEBEA. 



[CHAP. VIII. 



contain five quantities, a, d^ n, I, and >S', and consequently give 
rise to the following general problem, viz. : 

AtiT/ three of these Jive quantities being given, to determine the 
other two. 

This general problem gives rise to the ten following cases : — 



No. Given, f.^nknown 



Values nf the Unknown Quantities. 



a, c/, n 



L S I 



{n-l)d; S =^n[2a + (n — l)d]. 



a, d, I 



n, S 



— hi; a5^ 



2d 



a, d, S 



n, I 



7^= ^51 ' I = a -i- [n — l)d. 



a, n, S 



S, d 
d, I 



S= l?2 (a + ; ^ = 



l-a 
n- V 



^_ 2{S-an) ^ 2S_ 

n(n-l) ' n 



a, Z, S 



n, d 



2S _ ^_ (/+«)(/ -a) 



a-h/' 2o-(/ + a) 



d, «-, / 



tf, S 



a=: I - (n -1)6/; S = ^n [21 - {n - 1) d]. 



l,n,S 



a, I 



_ 2S — n {n — l)d 
a_ ~ -; 



2S-^n(n-l)d 
2n * 



d,l,S 



2l-{-dzt^{2i+d)^-8dS . . ^.^ 

""^ 2d ' a = l-{n-l)d. 



10 



n, I, S\ a, d 



2S , , 2(nI-S) 

a== I; d=—^ -^. 

Qi 11 {n — 1) 



177. From the formula 

I ^:z a -\- {n — \) d, 
we have, a=: I — {n — \)d \ that is, 

The first term of an increasing arithmetical progression., is eqiial 

to any following term, minus the product of the common difference 

by the number of preceding terms. 

178. From the same tormula, we also find 

I — a 

a —• ; that is, 

n — l 



CHAP. VIII. J ARITHMETICAL PROGRESSION. 239 

In any arithmetical progression, the common difference is equal 
to the last term minus the first term^ divided by the number of 
terms less one. 

If the last term is less than the first, the common diflerence 
will be negative, as it should be. 

EXAMPLES. 

1. The first term of a progression is 4 the last terrri 16, and 
the number of terms considered 5 : what is the common 
difference 1 \ 

> The formula 

Z-a . , 16-4 . 

cZ = -— ^ gives ,^ = -^^ = 3. 

2. The first term of a progression is 22, the last term 4, 
and the number of terms considered 10 : what is the common 
difference 1 Ans. — 2. 

179. By the aid of the last principle deduced, we can solve 
the following problem, viz. : 

To find a. number m of arithmetical means between two given 
numbers a and b. 

To solve this problem, it is first necessary to find the com- 
mon difference. Now, we may regard a as the first term of 
an arithmetical progression, b as the last term, and the required 
means as intermediate terms. The number of terms considered, 
of this progression, will be expressed by m + 2. 

Now, by substituting in the above formula, b for l^ and m H- 2 
for ?i, it becomes 

b — a ^ b — a 

'^= m + 2-l ' "" ^^Ti^+'V 

that is, the common difference of the required progression Is 
obtained by dividing the difference between the last and first 
terms by one more than the required number of means. 



240 ELEMEI>rTS OF ALGEBRA. [CHAP. VIIL 

Having obtained the common difference, form the second term 
of the progression, or the first arithmetical mean, by adding d, or 

-. to the first term a. The second mean is obtained by 

augmenting the first by d, &c. 

EXAMPLES. 

1. Find 3 arithmetical means between 2 and 18. The formula 

d z=z — — -, gives d z=z = 4 : 

hence, the progression is 

2 . 6 . 10 . 14 . 18. 



2. Find 12 arithmetical means between 77 and 12. The 
formula 

b -a . ^12-77 

hence, the progression is 

77 . 72 . 67 . 62 22 . 17 . 12. 

3. Find 9 arithmetical means and the series, between 75 
and 5. 

Ans. Progression 75 . 68 . 61 26 . 19 . 12 . 5. 

180. If the* same number of arithmetical means be inserted 
between the terms of a progression, taken two and two, these 
terms, and the arithmetical means together, will forn^ one and 
the same progression. 

For, let a . 6 . c . e . / . . . . be the proposed progression, 
and m the number of means to be inserted between a and i, 
b and c, c and ...... 

From what has just been said, the common difference of 
each partial progression will be expressed by 

b — a c — b e — c 

m+ 1' m -h 1' 7n + 1 * * * * 

f 

which are equal to each other, since, a, 6, c, . . . are in pro 

gression: therefore, the coniinon difference is the same m each 



CHAP. VIII.] ARITHMETICAL PROGRESSION. 241 

of the partial progressions; and since the last term of tiie first, 
forms the first terra of the second, &c., we may conclude thai 
all of these partial progressions form a single progression. 

GENERAL EXAMPLES. 

1. Find the sum of the first fifty terms of the progression 

2 . 9 . 16 . 23 . . . 

For the 50th term, we have 

Z = 2 + 49 X 7 = 345. 

50 
Hence, S={2^ 345) x — = 347 x 25 == 8675. 

2. Find the 100th term of the series 2 . 9 . 16 . 23 . . 

Ans. 695. 

3. Find the sum of 100 terms of the series 1.3.5.7.9... 

Ans. 10000. 

4. ITie greatest term considered is 70, the common difference 
8, and the number of terms 21 : what is the least term and 
the sum of the terms'? 

Ans. Least term 10; sum of terms 840. 

5. The first term of a decreasing arithmetical progression is 
10, the common difference is — ^, and the number of terms 
1\ : required the sum of the terms. Ans. 140. 

6. In a progression by differences, having given the commcm 
difference 6, the last term 185, and the sum of the terms 2945: 
find the first term, and the number of terms. 

Ans. First term =5; number of terms 31. 

7. Find 9 arithmetical means between each antecedent and 
consequent of the progression 2. 5. 8. 11. 14... 

Ans, rf = 0.3. 

8. Find the number of men contained in a triangular bat- 
talion, the first rank containing 1 man, the second 2, the third 
3, and so on to the n^^^ which contains n. In other worda, 

16 



242 ELEMENTS OF ALGEBRA. [CHAP.. VIII. 

find the expression for the sum of the natural numbers 1, 2, 
8, . . . from 1 to n, inclusively. , , ,. 

Ans. S = -, 

9. Find the sum of the first n terms of the progression of 
oneyen numbers 1, 3, 5, 7, 9 . . . Ans. S=zn^, 

10. One hundred stones being placed on the ground, in a 
straight line, at the distance of two yards from each other, how 
fer will a person travel who shall bring them one by one to 
a basket, placed at two yards from the first stone? 

Ans. 11 miles 840 yards. 

Of Ratio and Geometrical Proportion. 

181. The JRatio of one quantity to another, is the quotient 
which arises from dividing the second by the first. Thus, the 

ratio of a to b, is — . 
a 

182. Two quantities are said to be proportional, or in pro- 
portion, when their ratio remains the same, while the quantities 
themselves undergo changes of value. Thus, if the ratio of a 
to h remains the same, while a and h undergo changes of value, 
then a is said to be proportional to h. 

183» Four quantities are in proportion, when the ratio of the 

first to the second, is equal to the ratio of the third to the 

fourth. 

Thus, if 

h__d_ 

a c ' 

the quantities a, 6, c and c?, are said to be in proportion. We 
generally express that these quantities are proportional by writii.g 
them as follows : 

a : b '. : c : d. 

This algebraic expression is read, a is to 6, as c is* to d, 
4Uid is called a proportvm. 



CHAP. VIII.] GEOMETRICAL PROGRESSION" 243 

184. The quantities compared, are called terms of the pro- 
portion. 

Tlie first and fourth terl-is are called the extremes, the second 
and third are called the means ; the first and third are called 
antecedents, the second and fourth are called consequents, and the 
fourth is said to be a fourth proportional to the other three. 

If the second and third terms are the same, either of these 
is said to be a mean proportional between the other two. Thus, 
in the proportion 

a '. b : : b : c, 
5 is a mean proportional between a and c, and c is said to be 
a third proportional to a and b. 

185. Two quantities are reciprocally proportional when one is 
proportional to the reciprocal of the other. 

Geometrical Progression. 

186. A Geometrical Progression is a series of terms, each 
of which is derived from the preceding one, by multiplying it 
^y a constant quantity, called the ratio of the progression. 

If the ratio is greater than 1, each term is greater than i^ 
preceding one, and the progression is said to be increasing.. 

If the ratio is less than 1, each term is less than the pr*v 
ceding one, and the progression is said to be decreasing. 

Thus, 
... 3, 6, 12, 24, . . . &c., is an increasing progression. 

... 16, 8, 4, 2, 1, — , — , ... is a decreasing progressloiiL 

It may.be observed that a geometrical progression is a con- 
tinued proportion in which each term is a mean proportional 
between the preceding and succeeding terms. 

187. Let r designate the ratio of a geometrical progression^ 

a : b : c : d, . . . . &c. 
We deduce from the definition of a progression the follow 
ing equations : 

b = ar, c =: br =z ar"^, d = or =z ar^, e =: c?r = ar* . . j 



5H4 ELEMENTS OF ALGEBRA. LCHAP. VIIL 

sncl, Ji general, any term n, that is, one which has n — 1 terms 
before it, is expressed by ar"—^. 

Let / be this term ; we have the formula 
I = ar'^\ 
by means of which we can obtain any term without being 
obliged to find all the terms which precede it. That is, 

Any term of a geometrical progression is equal to the first term 
multiplied by the ratio raised to a power whose exponent denotes 
the number of preceding terms. 

EXAMPLES. 

1. Find the 5th term of the progression 

2 : 4 : 8 : 16, &c., 
m which the first term is 2, and the common ratio 2. 
5th term = 2 x 2* = 2 x 16 = 32. 

2. Find the 8th term of the progression 

2 : 6 : 18 : 54 . . . 
8th term = 2 x 3^ = 2 X 2187 = 4374. 
S. Find the 12th term of the progression 

64 : 16 : 4 : 1 : 4- . . 
4 

/ 1 \ii 43 1 1 

12th term == 64 (-)=—= -^ = ^^. 

188. We will now explain the method of determining the sum 
of n terms of the progression 

a '. b '. c '. d '. e '. f : . , , \ i : k '. I, 
of which the ratio is r. 

If we denote the sum of the series by S^ and the n'* term 
fey I we shall have 

S = a -\- ar -\- ar"^ . . . . -f ar"^^ -\- ar'^^^ 
If we multiply both members by r, we have 

Sr = ar ■+- ar^ -j- ar^ . . . -f- a^"""^ + o,r* ; 



CHAP. VIII.l GEOMETKICAL PROGRESSION. 245 

and by subtracting the first equation frcm the second, member 
from member, 

ar^ — a 

Sr — S = ar^ — a, whence, S = r- ; 

r — L 

substituting for ar", its value /r, we have 

Ir — a . 
S = r ; that IS, 

To obtain the sum of any number of terms of a progression 
by quotients, 

Multiply the last term by the ratio, subtract the Jirst term from 
this product^ and divide the remairider by the ratio diminished by 1, 

EXAMPLES. 

1. Find the sum of eight terms of the progression 

2 : 6 : 18 : 54 : 162 . . . : 4374. 

^ = ^1^=2^1?^=:^ = 6560. 
r — \ 2 

2. Find the sum of five terms of the progression 

2 : 4 : 8 : 16 : 32 ; . . . . 

r — \ 1 

3. Find the sum of ten terms of the progression 

2 : 6 : 18 : 54 : 162 ... 2 X 39 = 39366. 

Ans. 59048. 

4. What debt may be discharged in a year, or twelve months, 
by paying $1 the first month, $2 the second month, $4 the third 
month, and so on, each succeeding payment being double the 
last ; and what will be the last payment ? 

Ans. Debt, $4095 ; last payment, $2048. 

5. A gentleman married his daughter on New- Year's day, and 
gave her husband Is. toward her portion, and was t* double it 
on the first day of every month during the year : what was hei 
portion? Ans. £204 I5«, 



246 ELE^IENTS OF ALGEBKA. [CHAP. VTIL 

6. A man boiiglit 10 bushels of wheat on the condition that 
he should pay 1 cent for the first bushel, 3 for the second, 9 
for the third, and so on to the last : what did he pay for 
the last bushel, and for the ten bushels? 

Atis. Last bushel, |196 83; total cost, $295,24, 

189* When the progression is decreasing, we have r < 1 and 
f < a ; the above formula for the sum is then written under 
the form 

1 — r 
in order that both terms of the fraction may be positive. 
By substituting ar"~^ for I, in the expression for S, 

S = r-, or S=z— . 

r — 1 * 1 — r 

EXAMPLES. 

1. Find the sum of the first five terms of the progression 

32 : 16 : 8 : 4 : 2. 

32-2 X— ^. 

1 — r 1 1 

T "2" 

2. Find the sum of the first twelve terms of the progression 

1 1 



64 : 16 : 4 : 1 



4 65536* 



a_ fi — lr _ ^^ 65536^ 4 _ ^^^ ^5536 _ ^^ 6553 5 
'^ -r=y - ~£ - 3~ - ^^ ^ "196608* 

4 

We perceive that the principal difficulty consists in obtaining 
the numerical value of the last term, a tedious operation, even 
when the number of terms is not very, great. 

190i If in the formula , 

a(r- - 1) 



CKAO*. VIII.l GEOMETRICAL PROGRESSION. 247 

we make r — l, it reduces to 

<»■ 

This result sometimes indicates indetermination ; but it often 
arises from the existence of a conimon factor in both numerator 
and denominator of the fraction, which factor becomes 0, in con- 
sequence of a particular supposition. 

Such is the fact in the present case, since both terms of the 
fraction contain the factor r — 1, which becomes 0, for t4ie par- 
ticular supposition r =z 1. 

If we divide both terms of the fraction by this common factor, 
we shall find (Art. 60), 

S = ar''-'^ + ar""-^ + ar""-^ + .... -{• ar -{- a, 

in which, if we make ?■ = 1, we get 

S=za-{-a-{-a-\-a-\- +a= na. 

We ought to have obtained this result; for, under the suppo- 
sition made, each term of the progression became equal to a, 
and since there are n of them, their sum ^hould be na. 

191. From the* two formulas 

r^ ^^ — « 
I — ar^-^, and >S = - 



r-V 

several properties may be deduced. We shall consider only 
some of the most important. 
The first formula gives 

I ^ , «-i /T 

r"~^ = — , whence r= \ / — . 
a V a 

The expression 

furnishes the means for resolving the following problem, viz . 

To Jind m geometrical means between two given numbers a and 
b ; that is, to find a number m of means, which will form with a 
and h, considered as extremes, a geometrical progression. 



248 ELEMENTS OF ALGEBRA. LCHAP VIIL 

To find this series, it is onlj necessary to know the ratio. 
Now, the required number of means being m, the total number 
of terms considered, will be equal to m + 2. Moreover, we 
have I =^ b ; therefore, the value of r becomes 

r = \/ — ; that is. 

To find the ratio, divide ike second of the given numbers by the 
first; then extract that root of the quotient whose index is one 
greater, than the required riumber of means : 
Hence the progression is 

m+l r^ m+l r^ m+l H^ 



a 



EXAMPLES. 



1. To insert six geometrical means between the numbers 3 
and 384, we make m — Q, whence from the formula. 



V^=v^=^; 



hence, we deduce the progression 

3 : 6 : 12 : 24 : 48 : 96 : 192 : 384. 
2. Insert four geometrical means between the numbers 2- and 
486. The progression is 

2 : 6 : IS : 54 : 162 : 486. 

Remark. — When the same number of geometrical means are 
inserted between each two of the terms of a geomet.'ical pro- 
gressio]), all the progressions thus formed will, when jaken to- 
gether, constitute a single progression. 

Progressions having an infinite number of terms. 

192t Let there be the decreasing progression 
a : b : c : d : e '. f '. . . ., 
eontaining an infinite number of terms. The formula 

a — a?'" 

^ — "1 ^T' 

1 — r 



CHAP. VIII.] GEOMETRICAL PROGRESSION. 249 

lirhich expresses the sum of n terms, can be put under the form 

o = 



1 -r 



Now, since the progression is decreasing, r is a proper frac- 
tion, and r" is also a fraction, which diminishes as n increases. 
Therefore, the greater the number of teims we take, the more 

will X r^ diminish, and consequently, the nearer will the 

1 — r 

sum of these terms approximate to an equality with the first 
part of S ; that is, to . Finally, when n is taKen greater 

than any assignable number, or when 

a 

n = CO, then X r» 

1 — r 

will be less than any assignable number, or will become equal 
ko ; and the expression will represent the true value of 

the sum of all the terms of the series. Hence, 

The sum of the terms of a decreasing progression, in which the 
number of terms is infinite, is 

a 



S = 



1 - / 



This is, properly 'speaking, the limit to which the partial sums 
approach, as we take a greater number of terms of the pro- 
gression. The number of terms may be taken so great as to 

make the difference between the sum, and , as small as 

1 — r 

we please, and the difference will Duly become zero -v^hen the 

number of terms taken is infinite. 



EXAMPLES. 

1. Find the sum of 

_ 1 1 1 1 . 



250 ELEMENTS OF ALGEBRA. [CHAP. VIIL 

We have, for the sum of the terms, 



2. Again, take the progression 

1 -l.-L. -1.1.1. & 

We have S = -^— = ~^—- = 2. 

1 — r 1 

~"2" 

What is the error, in each example for ?i = 4, n = 5, n =: Q ? 

Indeterminate Co-efficients. 

193. An Identical Equation is one which is satisfied for any 
values that may be assigned to one or more of the quantities 
which enter it. It differs materially from an ordinary equation. 

The latter, when it contains but one unknown quantity, can 
only be satisfied for a limited number of values of that quan- 
tity, whilst the former is satisfied for any value whatever of 
the indeterminate quantity which enters it. 

It differs also from the indeterminate equation. Thus, if in 

the ordinary equation 

ax -{- b2/-\- cz +.c/ = 

values be assigned to x and y at pleasure, and corresponding 
values of z be deduced from the equation, these values taken 
together will satisfy the equation, and an infinite number of 
sets of values may be found which will satisfy it (Art. 88). 
But if in the equation 

ax -{- bi/ -{■ cz -\- d = 0, 
we impose the condition that it shall be ^tisfied for any 
valies of X, y and 0, taken at pleasure, it is then called an 
identical equation. 

194. A quantity is indeterminate when it admits of an infinite 
number of values. 

Let us assume the identical equati'^n, 

A^Bx-\- Cx'''-i-Dx^^ &i,Q = ^ .... (1), 



CHAP. VIII.] GEOMETRICAL PROGRESSION. 251 

in which the co-efficients, A, B, (7, i>, &;c., are entirely inde 
pendent of x. 

If we make x = in equation (1) all the term? containing 
X reduce to 0, and we find 

A = 0. 
Substituting this value of A in equation (1), and factoring, 
it becomes, 

x{B-\- Cx-\- Dx^ + &c.,) r= - - - - - (2), 
which may be satisfied by placing a; r= 0, or by placing 

B+ Cx + Bx^-i- &LG. = (3). 

The first supposition gives a common equation, satisfied only 
for X = 0. Hence, equation (2) can only be an identical equa- 
tion undpr a supposition which makes equation (3) an identical 
equation. 

If, now, we make a; = in equation (3), all the terms con- 
taining X will reduce to 0, and we find 

^ = 0. 
Substituting this value of B in equation (3), and factoring, 
we get 

x{C + Bx-\- &G.) = (4). 

In the same manner as before, we may show that (7=0, 
and so we may prove in succession that each of the co-efficients 
Z>, BJ, &c., is separately equal to : hence, 

In every identical equation, either member of which is 0, in- 
volving a single indeterminate quantity, the co-efficients of the 
different powers of this quantity are separately equal to 0. 

195. Let us next assume the identical equation 

a -\- bx -\- cx^ -\- &LG. = a' -f- b^x -f c'x^ -f- &c. 
By transposing all the terms into the first member, it may 
be placed under the form 

{a - a') -\- {b~b')x-\- [c — c') x"^ + &c. = 0. 
Now, from the principle just demonstrated, 

a — a' = 0, b — b' = 0, c — c' = 0, &c., &c. ; 
whence a = a\ b =- b' , c = c' , &c., &c. ; that is, 



^62 ELEMENTS OF ALGEBRA. ICHAP. VID. 

In an identical equation containing hut one indeterminate quan- 
tity, the co-efficients of the like powers of that quantity in the 
two members, are equal to each other. 

196. We may extend the principles just deduced to identical 
equations containing any number of indeterminate quantities. 

"For, let us assume that the equation 
a-f-bx-\-h'y-\- h"z -f &c. + cx^ + c'l/ + c"z^ + &c. + dt^ 
+ c?y + &c. = - . - (1), 
is satisfied independently of any values that may be assigned 
to re, y, 2, &;c. If we make all the indeterminate quantities 
xcept X equal to 0, equation (1), will reduce to 

a -\- hx -\- cx"^ -\- dx^ + &c. = ; 

whence, from the principle of article 194, 

a =: 0, ^> == 0, c = 0, c^ = 0, &c. 

If, now, we make all the arbitrary quantities except y equal 
to 0, equation (1) reduces to, 

a + &V + <^'y^ + ^'2/^ 4- &c. = ; 
whence, as before, 

a = 0, 6' = 0, c' = 0, d' = 0, (fee. 
and similarly we have 

5" = 0, c" = 0, &c. 

The principle here developed is called the principle of inde 
terminate co-efficients, not because the co-efficients are really 
indeterminate, for we have shown that they are separately 
equal to 0, but because they are co-efficients of indeterminate 
quantities. 

197. The principle of Indeterminate Co-efficients is much used 
ir developing algebraic expressions into seres. 

For example, let us endeavor to develop the expression, 

a 

a' -\-b'x' 

into a series arranged according to the ascending powers of »» 



CH/\.P. VIII.] GEOMETRICAL PROGRESSION. 253 

Let us assume a development of the proposed form, 
-r^-jr- = P + Qx -{- Bx"^ + Sx^ + ^^. - - -.(1), 

Qi —\~ O X 

in which P, Q^ R, &c., are independent of x, and depend upon 
o. a' and b' for their values. It is now required to find such 
values for P, Q, B, &c., as will make the development a true 
one for all values of x. 

By clearing of fractions and transposing all the terms into 
the first member, we have 

Fa' -{- Qa' x + Ra' x^ + &c. = 0. 
' -a ■\- Ph' -f Qh' &c. 

Since this equation is true for all values of a;, it is identi- 
cal, and from the principle of Art. 194, we have 
Pa' —az=, 0, qa' + Ph' = 0, Pa' + Qb' = 0, &c., &c. ; whence, 
P a _ Pb' ab' Qb' ab"^ ^ ^ 

a a a'^ a a'^ 

Substituting these values of P, ^, 7?, &c., in equation (1), 
it becomes 

a' ^b'x - a' a'2 "^ + a'3 ^ ^.4 ^ + <^c. - (2). 

Since we may pursue the same course of reasoning upon 
^ any like expression, we have for developing an algebraic ex 
pression into a series, the following 

RULE. 

I. Place the given expression equal to a development of the 
form P -\- Qx -\- Rx^ -\- ($:c., clear the resulting equation of frac- 
tions, and transpose all of the terms into the first member of 
the equation. 

II. Then place the co-efficients of the different powers :f the let» 
ier, with reference to which the series is arranged, separately equal 
to 0, and from these equations find the values of P, Q, R, d:c, 

III. Having found these values, substitute them for P, Q, R, dc, 
in the assumed development, and the result will be the develop- 
ment required. 



254 



ELEMENTS OF ALGEBKA. 



LCHAP. vin. 



1. Develop 



a — X 



EXAMPLES. 



into a series. 



Ans. l-\ 1-4.-1- &C. 



2. Develop 



3. Develop 



(a - xy 



1 + 2.r 



into a series. 



Ans. 



+ 



2x . Sx^ 



4.r3 



a^ a* a^ 



uito a series. 



1 - 3a: 

Ans. 1 + 5^ + 16x^ + 45a:^^ + lS5x* + &c. 

198. We have hitherto supposed the series to be arranged 
according to the ascending powers of the unknown quantity, 
commencing with the power, but all expressions cannot be 
developed according to this law. In such cases, the application 
of the rule gives rise to some absurdity. 

For example, if we apply the rule to develop 

shall have. 



Sx-x^' 



we 



:=F+ Qx-i-Bx^-\-&^o. - . . 

ox — x^ 

Clearing of fractions, and transposing, 

a;2 -f &c. = ; 



(1). 



— \+ZPx-^ZQ 
- P 
Whence, by the rule, 

-1=0, 3P = 0, 3^-P = 0,&c. 

Now, the first equation is absurd, since — 1 cannot equal 0. 
Hence, we conclude that the expression cannot be developed ao 
cording to the ascending powers of x, beginning at x^. 

We may, however, write the expression under the font 

— X ^ , and by the application of the rule, develop the facte 

x 3 — X 

, which gives 

Z — X 



l = T + ¥* + ^^'+§i^' + ^<^' 



CHAP. VIII.J RECURRING SERIES. 256 

whence, by substitution, 



Sx-x^ Sx ' 9 ■ 27 ' 81 

Since — is equal to Sx-^ (Art. 166), we see that the true devel- 
opment; contains a term with a negative exponent, and the sup- 
position made in equation (1) ought to have failed. 

Becurring Series, 

199. The development of fractions of the form — — 77-, &c., 

a'+ ox 

gives rise to the consideration of a kind of series, called recur- 
ring series. 

A RECURRING SERIES is ouc in which any term is equal to the 
algebraic sum of the products obtained by multiplying one or 
more of the preceding terms by certain fixed quantities. 

These fixed quantities, taken in their proper order, constitute 
what is called the scale of the series. 

200. If we examine the development 

a a ah' ab'"^ , ab'^ , 

a -{-bx a' a 2 ^/s gjA. 

we shall see, that each term is formed by multiplying the pre- 
ceding one by -x. This is called a recurring series of the 

first order ^ because the scale of the series contains but one 
term. 

The expression — ~x is the scale of the series, and the ex 

. b' . 
■ pression ^ is called the scale of the co-efficients. 

It may be remarked, that a geometrical progression is a recur 
ring series of the first order. 

201. Let it be required to develop the expression 

a -\- hx 



7! 4- h'x 4- c'x^ 



into a series. 



)6 


ELEMENTS 


OF ALGEBRA. [CHAP. VIIL 1 


Assume 


a -{- bx 


- = P + ^a; + J?a;2 + ^a;3 -f- &C. " 


a' + b'x + c'x 


Clearing of fractions, and transposing, we get 


Pa' 


+ Qa' 


x-\-Pa' 


x"^ + Sa' 


a;3 + &c. = 0. 


— a 


+ Pb' 


+ Qb' 


-\-Pb' 


1 


-b 


-\-Pc' 


-i-Qc' 


i 


Therefore, we have 


^ 


Pa' - a = 0, 


or, P = ^,, 


• 
Qa' + Pb' - b=0, 


-' «=-4'^+i^ 


Pa' + Qy -i- P(/=0, 


-' '' = -v'^->' 


Sa' + Pb' + Q(/ = (i, 


- ^ = -^^-^« = 


&;c., 


&c., 






&c., &;c.; 



from which we see that, commencing at the third, each co-effi- 
cient is formed by multiplying the two which precede it, re. 

spectively, by j and 7, viz., that which immediately 

pi-ecedes the required co-efficient by ^, that which precedes 



it two terms by 
ducts. Hence, 



(/ 



J, and taking the algebraic sum of the pro 



\ a'' a') 



is the scale of the co-efficients. 

From this law of formation of the co-efficients, it follows thai 
the third term, and every succeeding one, is formed by multi- 

plying the one that next precedes it by jx^ and the second 

preceding one by ; a;^, and then taking the algebraic sum of 

these products : hence, 






is the scales of the series. 



CHAP. VllI.J BINOMIAL THEOREM. 257 

This scale contains ' two terms, and the series is called a re- 
curring series of the second qrder. In general, the order of a 
recurring series is denoted by the number of terms in the scale 
o£ the series. 

The development of the fraction 

a -i- bx -^ cx^ 
a' ^h'x^ex^-\-d'x^' 

gives rise to a recurring series of the third order, the scale of 
which is, 

and, in general, the development of 

o, -\- hx -{■ cx^ ■\- . . . A;a:"— ' 
a' -\-h'x^cfx^ -\- . . . h'x'' ' 

ejives a recurring series of the n^^ order ^ the scale of which is 
\ of a' of I 

General demonstration of the BinoTiiial Theorem. 

202. It has been shown (Art. 60), that any expression of the 
form z^ — y^^ is exactly divisible by z — y, when m is a po«i^iv<» 
whole number, giving, 

gWl nifll 

— =r z'^-^ 4- z'^-'^y + z''nr-^y'^ + . . . . -f y*"-!. 

The number of terms in the quotient is equal to m, and if 
we suppose 2 = 2/, each term will become ^""-^ ; hence, 



z~y )y = z~ 



The notation employed in the first member, simply indicates 
what the quantity within the parenthesis becomes when we make 
y =z. 

We now propose to show that this form is true when tn \s 



fractional and when H is negative. 



17 






268 ELEMENTS OF ALGEBRA. LCHAP. VIIL 

P 

First, suppose m fractional, and equdtl to — . 

Make z^ = v, whence zi =vp and z = vi\ 

and y^z=zu, whence yi^^u^ and y =u9 , 



hence, 



2? — y? vT? — uP V — u 

2 — y 1)1 — Ul 1)1 — Ul' 

V — u 
If now, we .suppose y •=^z, we have v = w, and since p and q 
are positive whole numbers, we have 



p p 



(vP — uP\ 
V — U Jv: 



,jy-l ^ ^ P 



s q — 1/9 I \v — u /v = H pvP-^ p p -^-i 

^ \ — ^ ' _ ^ — _ 6LyP—q = ±-z9 



J /Vl — U9\ 



qvl- 



Second, suppose m negative, and either entire or fractional. 

By observing that 

— ^-OT y-m y^ ^^m — 2/"*) = z~^ — y~^i 
we have, 

g m ir~'^ s"* ~— I/"* 

— = — 2-^ y-^ X —. 

z — y z — y 

If, now, we make the supposition that y =■ z, the first factoi 
of the second member reduces to — z-"^^, and the second fac- 
tor, from the principles just demonstrated, reduces to m^'^- ; 
hence, 

\ Z-y Jy=z 

We conclude, therefore, that the form is general. 

203. By the aid of the principles demonstrated in the last 
article, we are able to deduce a formula for the develop- 
ment of 

{x + a)^, 

when the exponent m is positive or negative, entire or fractional 
Let us assume the equation, 

(l-^z)'^ = F-i- Qz + Rz^ + Sz^ + &;c. - - (1), 



CHAP. VIII.J BINOMIAL THEOREM. 25^ 

in which, P, Q, i?, &c., are independent of z, and depend upon 
1 and m for their values. It is required to find such values 
for them as will make the assumed development true for every 
possible value of z. 

It] in equation (1) we make z =: 0, we have 

P=: 1. 
Substituting this value for P, equation (1) becomes, 

(1 + g)'» = 1 + ^2 + Mz^ + Sz^ 4- &c. - - - (2). 
Equation (2) being true for all values of 0, let us make e = y^ 
whence, 

(1+ y)- =z 1 + ^y + P^/2 4- ^2/3 4. &c. - - - (3). 
Subtracting equation (3) from (2), member from member, and 
dividing the first member by (1 + 0) — (1 -}- y), and the second 
member by its equal z — ?/, we have, 

(l+it-_ll+ ^1! = «i^ +iefl_-zl! + ^ii^' + &e. . (4). 

(lH-2)-(l+y) ^z-7j z-y z-y ^ 

If, now, we make 1 + = 1 -f ?/, whence z-=Ly^ the first 
member of equation (4), from previous principles, becomes 
7?i(l+2)'"-\ and the quotients in the second member become 
respectively, 

C-^3 =^fe~l =^<^ =3.^..c.^c. 

Substituting these results in equation (4) we have, 

m (1 + zy-^ = ^ + 2P^ -I- ZSz' + 4 7^23 4- &c. - - - (5). 

Multiplying both members of equation (5) by (1 + 2;), w^e find, 



m(l +0)'"= ^ + 2P 
+ Q 



^ + 3>S' 
+ 2P 



^2-|-4r 
+ 35 



^3 4- &c. - - - (6). 



If we multiply both members of equation (2) by m, we havo 
m (1 -^ zY — m^rnQz^ mRz^ + mSz^ + m^s* + &c. (7). 

The second members of equations (6) and (7) are equal to 
each other, since the first members are the same ; hence, we 
have the equation. 



m-\'mqz~\-mRz^-\-mSz^^h(^.^q^-1R\z^-%S 

-L- q, 42P 



z'^^T 
+35 



23-f &c-(8) 



260 ELEMENTS OF ALaEBBA. [CHAP. VIII 

This equation being identical, we have, (Art. 195), 
C=m, . - or, . . Q=—, 

%B+Q = mQ,. or, - - ^^ ^(^^-1) . 

%S+2R = .^R, - or, - - ^ ^ ^(^ - I H^ - 2) . 

^T+ZS^mS, . or, - ^ ^ m(m - 1) (m - 2^ (m-- 3)^ 

1 , Z . o . ^ 

&c., &c., &c. 

SuKstituting these values in equation (2), we obtain 

If now, in the last equation, we write — for 0, and then mul. 
tiply both members by rr"*, we shall have, 

, . m{m—\) ^ - , m(m — l)(m— 2) „ 

+ &c. . . (10). 

Hence, we conclude, since this formula is identical with that 
deduced in Art. 136, that the form of the development of (a;-f a)*" 
will be the same, whether m is i^osiiive or negative^ entire or 
fractional. 

It is plain that the number of terms of the development, when 
m is either fractional or negative, will be infinite. 

Ap2jlica*mis of the Binomial Formula. 

204. If in the formula (x -f- a)« = 

(a . m—\ a^ , m — 1 m — 2 a^ \ 



CHAP, yill.] BINOMIAL THEOREM - 261 

we mako m = — , it becomes (a; -f- «) " or ^ a; -i- a — 

or, reducing, «/ a; + a = 

1/, la 1 n — 1 a'^ 1 ^ — 1 2w — 1 tt3 \ 

\ n' X n ' 2/i ' a;^ n ' 2n ' Sn ' x^ ' * * / 

The fifth term, within the parenthesis, can be found by mul- 
tiplying the fourth by — and by — , then changing the sign 

rlW X 

of the resillt, and so on. 

205. The formula just deduced may be used to find an appro*, 
imate root of a number. Let it be required to find, by means 
of it, the cube root of 31. 

The greatest perfect cube in 31 is 27^ Let x = 27 and a = 4oi 
making these substitutions in the formula, and putting 3 in the 
place of n, it becomes 

V - \ V" 3 • 27* 3 • 3 ' 729 "^ 3 * 3 • 9 • 19683 

_1 111 256 \ 

3 • 3 • 9 • 3 * 531441 + ^' j 

or, by reducing, 

3 /oT-_ Q . ± _ Jl. . g^Q _ 2560 

V "^ - "^27 2187 "*" 531441 43046721 "^ ^' 

Whence, ^/3T = 3 . 14138, which, as we shall show presently, 
is exact to within less than .00001. 

We may, in like manner, treat all similar cases : hence, foi 
extracting any root, approximatively, by the binomial formula, 
we have th^ following 

RULE. 

Find the perfect power of the degree indicated, which is nearer 
to the given number, and place this in the formula for x. Sub- 
tract this power from the given number, and substitute this differ- 
ence, which will often be negative, in the formula for a. Perform 
the operations indicated^ and the result will be the required root. 



262 ELEMENTS OF ALGEBRA. [CHAP. VTII. 



EXAMPLES. 



X/ 1 \ » 



2. V^0~= (32 - 2)* rzr 32^ A - 3^) = 1.9744. 

3 

3. \/'39~= (32 4- 7)*" = 32^ /l + ^) = 2.0807. 

1 
4; y 108 = (128 - 20)^ = 128M 1 - 09 ) = 1.95204. 

£06* When the terms of a series go on decreasing in value, 
the series is called a decreasing series ; and when they go on 
increasing in value, it is called an increasing series. 

A converging series is one in which the greater the number 
of terms taken, the nearer will their sum approximate to a 
fixed value, which is the true sum of the series. When the 
terms of " a decreasing and converging series are alternately 
positive and negative, as in the firoc example above, we can 
determine the degree of approximation when we take «the sum 
of a limited number of terms for the true sum of the series. 

For, let a — b-\-c — d-{-e — /+ . . ., &c., be a decreasing 
series, b, c, d, . . . being positive quantities, and let x denote 
the true sum of this series. Then, if n denote the number of 
terms taken, the value of x will be found between the sums 
of n and n -{- 1 terms. 

For, take any two consecutive sums, 

C'-b-i-c — d-\-e — /, * and a — b -\- c — d -{- e — /+ g. 

In the first, the terms which follow — /, are 
+ g-h, + A: - / -f- . . ; 
but, since the series is decreasing, the terms g — h, k — I . , 
&c., are positive ; therefore, in order to obtain the complete 
value of X, a positive number must be added to the sura 
ff — l^C'-d-{-e—f. Hence, we have 

a — 6-fc — a?4-e— /■<a;. 



CHAP. VIII.J BINOMIAL FORMULA. 263 

In the second sum, the terms which follow + g^ are - k 
-\- k — I -\- m . . . . Now, — A -f A', — I -{- m . . . &c., are 
negative ; therefore, in order to obtain the sum of the series, 
a negative quantity must be added tc 

a — b-\-c-~d-{-e—f+g', 

or, in other words, it is necessary to diminish it. Consequently, 

a — b + c — d-i-e —f+ g > x. 

Therefore, x is comprehended between the sums of the first 
n and the first n -\- \ terms. 

But the difference between these two sums is, equal to g\ and 
riince X is comprised between them, g must be greater than 
the difference between x and either of them ; hence, the error 
committed by taking the sum of n terms, a — b -\- c — d -\- e — f, 
of the series, for the sum of the series is numerically less than 
the following term. 

207. The binomial formula serves also to develop algebraic 
expressions into series. 

EXAMPLES, 

1. To develop the expression , we have, 

[n the binomial formula, make ?w == — 1, a; = I, and a =j — t.^ 

and it becomes 

(1 _ ,)-! = 1 » 1 . (_ ^) ^ 1 . ulni . (__ ,)2 

2 3 ^ ^ 

o>, performing the operations indicated, we find for the de- 
velopment, 

= (1 - z)"^ = I + z -{-z^ -{- z^ -^ Z^-{- &LG. 

L ~~ Z 

We might have obtained this result, by applying the rule 
for division. 



264 ELEMENTS OF ALGEBRA. lCHAP. VUl 

2. Again take the expression, 

• (n^ or 2(1 -.)-3. 

Substituting in the binomial formula — 3 for m, 1 fcr x, 
and — z for a, it becomes, 

S-l-3-2. ,, . 

- 3.—^-. — ^—-i-^y - &c. 

Performing the indicated operations and multiplying by 2, 
\fe find 

2 



(1-^y 



2 (1 4- 32 + 6^2 4- 10^3 + 152* + &c.). 



3. To develop the expression ^^2z~z^ we first place it 
Mnder the form ^^^^xfl — ttj - ^Y ^^® application of the 



binomial formula, we find 



1 1 1 2 ^ 3 

-^ 6 ^ 36 ^ 648 ^ • • • ' 

hence, 3^2z - z^ z= ^^^^yl - — z - — z^ - -^z^ -, &cj 

4. Develop the expression -; -— = (a + ^)~^ hito a series 

r2 

5. Develop into a series. 

r -\- X 

/^2 /v3 /jj4 

^ns. r — a; 4 H 5, &c. 

0,2 _L .-^2 

(). Develop the square root of -r '— into a series. 



^'"•^ + 7^ + 2;r« + 27e'*'«- 

7. Develop the cube root of ^-- -— into a series. 






CHAP. YIII.J SUMMATION OF SERIES. 2^5 

Bummation of Series. 

208. The Summation of a Series, is the operation of finding 
an expression for the sum of any number of terms. Many- 
useful series may be summed by the aid of two auxiliary series. 

Let there be a ffiven series, whose terms may be derived from 
the expression — — -- — -, by giving to ^ a fixed value, and then 
attributing suitable values to q and n. 

Let there be two auxiliary series formed from the expressions 
— and — ; — , so that the values of p, q. and n. shall be the 
same as in the corresponding terms of the first series. 

It can easily be shown that any term of the first series is 

equal to — multiplied by the excess of the corresponding term 

in the second series, over that in the third. 
For if we take the expression 

p\n n -{■ •pf 
and perform the operations indicated, we shall get the expression, 

■ hence, we have 



n{n -4- p) ' 

p\n n -{■ p)} ^ 



n(n + p) 

which was to be proved. 

It follows, therefore, that ike sum of any number of terms oj 

the first series, is equal to — multiplied by the excess of the sum 

of the corresponding terms in the second series, over that of the 
corres^wnding terms in the third series. 

Whenever, therefore, we can find this last difierence, it ia 
always possible to sum the given series. 



266 ELEMENTS OF ALGEBRA. [CHAP. VIIL 



EXAMPLES. 

1. Requirec the sum of n terms of the series 

_L+_L + _L+J_ + &e. • 

1.2^ 2.3 ^ 3.4 ^4.5^ 

Comparing the terms of this series with the expression 

g 

n{n + pY 

we see that making i? = 1, ? = 1, and w = 1, 2, 3, 4, &c., in 
succession, will produce the given series. 

The two corresponding auxiliary series, to n terms, are 

■H-i-T+ h 

„a i+i+±+ i+ ■ 



234 n n-{-l 

Tlie difference between the sums of n terms of the first and 
second auxiliary series is 

, or, if we denote the sum 



n+ 1 
of n terms of the given series by S^ we have, 

5=1 1 



71+1 

If the number )f terms is infinite n =: co and 

^ = 1. 
2. Eequired the sum of n terms of the series 

O + 375 + 5?r + 779 + 97ri + *'''•' 

If we compare the terms of this series with the expression 

we see that p =i 2, q = I, and n = 1, 3, 5, 7, &c., in suc- 
cession. 



CHAP. VIII.J - SUMMATION OF SERIES. 267 

The two auxiliary series, to n terms, are, 





-4-4 


+i+... 


1 1 . 




' 2» - 1' 


AUd 


4-f 


+1+.., 


1 1 1 1 


■ ■ ■ ' 2»i - 1 ' 2« + 1 


hence, 


as before, 







^=W~2;rTi)- 



If « = 00, we find aS^ = — . 

3. Required the sum of n terms of the series 

Here jd = 3, q = 1-, n = 1, 2, 3, 4, &c. 

The two auxiliary series, to n terms, are, 

1+i +^+-^+ ^ 



4 5 n n-\- \ n + ^ 71+3' 

hence, ^^ -i(l +i + 1- - -i-^ - -i-^ 

If 7l=G0, aS =— . 

4. Required the sum of the series 

4 4 4 4 4 

1.5^5.9^ 9 . 13 ^ 13 . 17 ^ 17 . 21 ^ 

5. Find the sum of n terms of the series, 

3.5 5.7 "^7.9 9.11 "^11.13 ^^r ' ' " 

Here i? = 2, g = 2, - 3, +4, - 5, -f- 6, &t5. 

w=: 3, 5, 7, 9. 11, &c. 



ELEMENTS OF ALGEBRA. LCHAP. VUI. 



The two auxiliary series are, 



3 5 ' 7 9 ' ' 2n-\-l 

,2 3 4 71 , ■;. + 1 



5 7 ' 9 2/t -h 1 2/1 -h 3 ' 

aence. ^^ i(| - ,^) - | d - 1 + 1 - • • • - D- 

If n is eve/i, the upper sign is used, and the quantity in 
the last parenthesis becomes 4-1, hi which case 

1 /2 Ti + n 1 _ 1 / 1 ^L±i\ 

2 \ 3 "*" 2r. + 3/ 2 2 V 3 ' 2;i -f 3/* 
If n is odd, the lower sign is used, and the quantity in tt 
last parenthesis becomes 0, in which case 

1 /2 7l+l \ 

2 V 3 2^ + 3/ 

If in either formula we make 

n-{-l 1+"^ . 1 ^ ^ 1 

^^ = ^'2^^ = 3- ^^^^"^^^ "2' ^^^ ^=12- 

6. Find the sum of n terms of the series, 

1.3 2. 4*^3. 5 4.6' ^* 
Here, ^ = 2, g = 1, - 1, +1, - 1,+ 1, - 1, &c 

71= I, 2, 3, 4, &c. 
The two auxiliary series are, 

2 3 4 o 6 71 

^3 4 '5 6^ ^71 n+1 n-i i 

whence, >Sf = — ( — hF , , ± — --A. 

2\2 71 + 1 7i + 2/ 

If n = 00, we find S = —-. 

4 



CHAP. VIII.] METHOD BY DIFFERENCES. 269 

Of the Method hy Differences, 

209. Let a, 6, c, c? . . . . dec, represent the successive terms 
of a series formed according to any fixed law ; then if each 
term be subtracted from the succeeding one, the several re 
mainders will form a new series called the first order of dif- 
ferences. If we subtract each term of this series from the 
succeeding one, we shall form another series called the second 
order of differences^ and so on, as exhibited in the annexed 
table. 

6 — a, c— 6, c?— c, e — c?, &c., 1st. 

c— 26+a, d—2c + 6, e~2d-{-c, &c., 2d. 

d—Zc-\-Zh—a, e— 3cZ+ 3c — 5,&c., 3d. 

e — 4(^ + 6c — 4^ + a, &c., 4th. 

l?^ now, we designate the first terms of the first, second. 
third, &;c. orders of differences, by c?i, c/j, d^^ d^, &;c., we shall 
have, 

di = b — a, whence b=: a -h c?i, 

di = c — 2b -\- a^ whence c = a -\- 2di -\- c?2, 

d^ = d — 3c 4- 35 — a, whence dz= a -\- od^ + Sd^ -f- c?3, 

d^ = e — 4d -{- 6c — 4b -\- a, whence e =^ a -\- Ad^ -\- 6d^ -f 4:d^ + d^, 
&c. &c. &c. &c. 

And if we designate the term of the series which has n 
terms before it, by T, we shall find, by a continuation of 
the above process, 

^ .■(»-i)(.-y(.-s) ^^^^^ - - - (1). 

* 
This formula enables us to find the {n -\- 1)*^ term of a 

series when wc know the first terms of the successive orders 

jf differences. 



270 ELEMENTS OF ALG^EBRA. [CHAP. YITI. 

210. To find an expression for the sum of n terms of the 
series a, b, c, &;c., let us take the series 

0, a, a-{- b, a-\- b -{- c, a + 6 + c + c?, &c. - - - , (2) 
The first order of differences is evidently 

a, b, c, d, &c. Jt^^^ 

Now, it is obvious that the sum of 9i terms of the series (3), 
is equal to the (n + 1)*'* term of the series (2). 

But the first term of the first order of differences in series (2) 
is a; the first term of the second order of differences is the 
same as d^ in equation (1). The first term of the third order 
of differences is equal to d.^, and so on. 

Hence, making these changes in formula (1), and denoting the 
sum of n terms by S, we have, 

^-''''-^ -772""^^ + 17273 '^^+ 1.2.3.4~~" ^^ 

-r &c (4). 

When all of the terms of any order of differences become 
equal, the terms of all succeeding orders of differences are 0, 
and formulas (1) and (4) give exact res^ilts. When there' are 
no orders of differences, whose terms become equal, then for- 
mulas do not give exact results, biit approxiinations more or less 
exact according to the number of terms used. 

EXAM1»LES. 

1. Find the sum of ?i terms of the series 1.2, 2.3, 3.4, 
4 . 5, &c. 

Series, 1.2, 2.3, 3.4, 4.5. 5 . 6, &c. 

1st order of differences, 4, 6, 8, 10, &C. 

2d order of differences, 2, 2, 2, &c. 

3d order of differences, 0, 0. 

Hence, we have, a = 2, d^ — 4, d^ = 2, c4, d^, &c., equal 
to 0. 



CHAP. VIII.. METHOD BY EIFFERENCES. 271 

Substituting these values for g, c^j, d^^ &c., in formula (4), 
we find, 

g^a« + !ii^X4+ "("-/H';-^) x2; 
whence, ^ ^ »(» + 1 ) («_+_2)^ 

2. Find the sum of n terms of the series 1.2.3, 2.3.4, 
S.4.5, 4.5.6, &c. 

1st order of differences, . 18, 36, 60, 90, 126, &c. 

2d order of differences, 18, 24, 30, 36, &c. 

3d" order of differences, 6, 6, 6, &c. 

4th order of differences, 0, 0. &c. 

We find a — %, d^ = 18, d^ = 18, d^ =6, d^ = 0, &c. 

Substituting in equation (4), and reducing, we find, 

3. Find the sum of n terms of the series 1, 1+2, 1+2H-3, 
1 4- 2 + 3 + 4, &c. 

Series, 1, 3, 6, 10, 15, 21. 

1st order of differences, 2, 3, 4, 5, 6. 

2d order of differences, 1, 1, 1, 1. 

3d order of differences, 0, 0, 0, 

a = 1, di = 2, c?2 = 1, c?3 = 0, d^ = 0, &c. ; 

hence S -n I ^^^'^^ 2 1 ^(^-1) C^ -^) _ ^^ + ^^^ "^ ^^ . 
hence, ,b _7i -f ^^ .2+ ^-^^ _ ^^^ , 

- . ^ n(7i 4- 1) (n + 2) 
or, reducmg, S = -^ — ^23 

4. Find the sum of n terms of the series 1^, 2^, 3^, 4^, 5^, &c. 

We. find, a = 1, d,=:S, d, = 2, d^ = 0, d^ = 0, &c., &c. 

Substituting these values in formula (4), and reducing, we find, 

^ n{n -\- \) {2n -{- I) 
1.2 3* 



272 



ELEMENTS OF ALGEBRA. 



[chap: viil 



5. Find the sum of n terms of the series, 

1 . (7?i -f 1), 2{m-^ 2), 3 (wi + 3), 4 (m + 4), &c. 

We find, a = 771 -\- 1, d^ ^= m -\- S, d^ = 2, d^ = 0, &c. ; 

/ ,N n.(n — l), ^. n.(n—l)(n—2) 

whence, S =n {m -\- I) -1^ -^ — ^ (7?i + 3) + -^ — 23 ^ * 



or. 



S = 



n . {n + 1).{1 -\-2n-\- Sm) 



Of Filing Balls. 

The last three formulas deduced, are of practical appli- 
cation in determining the number of balls in different shaped 
piles. 

First, in the Triangular Pile. 

211. A triangular pile is formed of succces- 
sive triangular layers, such that the number 
of shot in each side of the layers, decreases 
continuously by 1 to the single shot at the 
top. The number of balls in a complete tri- 
angular pile is evidently equal to the sum 
of the series 1, 1+2, l-f2-f-3, 1+2 + 3 
+ 4, &;c. to 1 + 2 +...+«, ?i denoting the number of balls 
on one side of the base. 

But from example 3d, last article, we find the sum of n 
terms of the series, 

n{n-{-\){n + 2) 




1.2.3 

Second, in the Square Pile. 
212. The square pile is formed, 
ft3 shown in the figure. The num- 
ber of balls in the top layer is 1 ; 

"the number in the second layer is 
denoted by 2^ ; in the next, by 3^, 
and so on. Hence, the number of 
balls in a pile of n layers, is equal 
to the sum of the series, P, 2- 3^, 



(!)• 







CHAP. VIIL] 



PILING BALLS. 



273 



tc, n^, which we see, from example 4th of the last article, is 



S^ 



n.(?i-]-l).{2n+ 1) 



1.2.3 

Third, in the Oblong Pile. 



(2). 



m^Am^^BW^0&^9M 



U.Am.jmM.'m m 



^Kmmmsmmmmm^m^ 



213. The complete oblong pile has (m + 1) balls in thft 
upper layer, 2.(m + 2) in the next layer, 3 (m + 3) in tha 
third, and so on : hence, the number of balls in the complete 
pile, is given by the formula deduced in example 5th of the 
preceding article. 



n.('n + 1).(1 4-2n + 3m) 
'^^ 1 . 2 .~3 



(3). 



214. If any of these piles is incomplete, compute the nuiiv 
ber of balls that it would contain if complete, and the number 
that would be required to cpmplete it ; the excess of the for- 
mer over the latter, will be the number of balls in the pile. 

The formulas (1), (2) and (3) may be written, 



triangular, 

square, 

rectangular, 

Now, since 






(1); 
(a); 



S = J "^" ^ ^^ {(n+m) + {n+m) + (m+l)y{3). 

— ^-- is tne number of balls in the tri- 

angular face of each pile, and the next factor, the number of balls 
in the longest line of the base, plus the number in the side 
of the base opposite, plus the parallel top row, we have the 
following 

18 



274 ELEMENTS OF ALGEBRA. LCHAP. VIU. 

RULE. 

Add to the number of balls in the longest line oj the base the 
number in the parallel side opposite^ and also the number in the 
top parallel row ; then multiply this sum by one-third the number 
in the triangular face ; the product will be the number of balls in 
the pile. 

EXAMPLES. « 

* 

1 . How many balls in a triangular pile of 15 courses 1 

Ans. 680. 

2. How many balls in a square pile of 14 courses 1 and how 
many will remain after 5 courses are removed'? 

Ans. 1015 and 960. 

3. In an oblong pile, the length and breadth at bottom are 
respectively 60 and 30 : how many balls does it contain % 

Ans. 23405. 

4. In an incomplete oblong pile, the length and breadth 
at bottom are respectively 46 and 20, and the length and 
breadth at top 35 and 9 : how many balls does it contain 1 

Ans. 7190. 

5. How many balls in an incomplete triangular pile, the num 
ber of balls in each side of the lower course being 20, and 
in each side of the upper, 10? 

6. How many balls in an incomplete square pile, the number 
in each side of the lower course being 15, and in each side 
of the upper course 6 ? 

7. How many balls in an incomplete oblong pile, the num- 
bers in the lower courses being 92 and 40 ; and the numbers 
in the x)rresponding top c^irses being 70 and 181 



CHAPIER IX. 

CONTINUED FRACTIONS — EXPONENTIAL QUANTITIES LOGARITHMS, iMD 

FORMULAS FOR INTEREST. 

215. Every expression of the form 



3 


1 


1 


a+1^ 
b 


a+1^ 
6+1 


a+1' 
6+1 




c 


c+1 
d 



in which a, 6, c, c?, &c., are positive whole numbers, is called a 

continued fraction : hence, 

A CONTINUED FRACTION Jias 1 fov its numerator, and for its de- 
nominator, a whole number plus a fraction, which has 1 for its 
numerator and for its denominator a whole number plus a frac- 
tion, and so on. 

216i The resolution of equations of the form 

a' =:h, 

gives rise to continued fractions. 

Suppose, for example, a = 8, 6 = 32. We then have 

s"" = 32, 
ill which a; > 1 and less than 2. Make 

a: = 1 + 1, 

y 



276 ELEMENTS OF ALGEBRA. LCHAP. LX. 

m which 2l > 1, and the proposed equation becomes 

32 = 8 ^=8x8"; whence, 

8^ = 4, and consequently, 8 = 4. 
It is plain, that the value of y lies between 1 and 2. Suppose 









\ 


y=i + 7, 


and we 


have. 






8 = 4 ^=4x4^; 


Lence, 








4"^' = 2, and 4 = 2*, or z = 2. 


But, 






y 


-+1=-M> 


and 




X 


= 


1+1-1+ 1 -1+2 -^. 


l+y-l+,^l-l+ 3 -3' 



and this value will satisfy the proposed equation. 

For, 8* = 3y^ = 3^(2^ = \fWf = ^ = 32. 

21 7i If we apply a similar process to the equation 
' (10)' = 200, 
we shall f^nd 

« = 2 4--; y = 3 + li 2-3 + ~. 

Since 200 is not an exact power, x cannot be exactly ex- 
pressed either by a whole number or a fraction: hence, the 
value of X will be incommensurable with 1, and the contmucd 
fraction will not terminate, but wiU be of the form 

a: = 2 4--=2-|-? r- = 2+ ^ 



' 3 + i 3+1 



3.1 

u 4" &;c. 



CHAP. IX. 1 CONTlNtTED FRACTIONS. 277 

218» Vulgar fractions may also be placed under the form of 

continued fractions. 

65 
Let us take, for example, the fraction -r-rr^ and divide both 
' ^ 149 

its terms by the numerator 65, the value of the fraction will 

not be cnanged, and we shall have • 

^5 _ _L 

149 ~" j49' 

65 

^ .,-,.. . 65 1 . 

or enectmg the division, —-— i== — . 

^65 

19 

Now, if we neglect the fractional part, — , of the denomina 

oo 

toi, we shall obtain •«— - for an approximate value of the given 

fraction. But this value will ^e too large, since the denominoh 
tor used is too small. 

If, on the contrary, instead of neglecting the part — , we 

were to replace it by 1, the approximate value would be — -^ 

o 

which would be too small, since the denominator 3 is too 

large. Hence, 

1 ^ 65 ., 1 ^ 65 

Y>-149 ^"^ y<l49' 

therefore the value of the fraction is comprised between — and — ^ 

<i> 3 

If we wish a nearer approximation, it is only necessary to 

operate on the fraction -— as we did on the given fraction — ^, 
6o ^ 149^ 

find we obtain, 



hence, 



19 _ 1 

^19 
65 1 



149 ^ 1 



^+i 



278 ELEMENTS OF ALGEBRA. 'CHAP. IX. 

Q 

If, now, we neglect the part — , the denominator 3 will "be less 
than the true denominator, and — will be larger than the num 

o 

ber which ought to be added to 2 ; hence, 1 divided b j 2 + — - 

will be less than the true value of the fraction ; that is, if we 
stop after the first reduction and omit the last fraction, the 
result will be too great ; if at the second, it will be too small, &c. ; 
and, generally. 

If we stop at an odd reduction^ and neglect the fractional part 
that comes after ^ the result will be too great ; hut if we stop at 
an even reduction, and neglect the fractional part that follows, the 
result will be too small. 



^ a 


' b' c ' •' 


a continued fraction, are called 


integral fractions 


The fractions, 




1 1 

J. 1' 


J, &c. 
a + 

b^L 



are caUed approximating fractions, because each gives, in succes- 
sion, a nearer approximation to the true value of the fraction : 
bonce, 

An approximating fraction is the result obtained by stopping 
at any integral fraction, and neglecting all that come after. 

If we stop at the first integral fraction, the resulting approxi- 
mating fraction is said to be of the first order ; if at the second 
integral fraction, the resulting approximating fraction is of the 
second order, and so on. 

When there is a finite number of integral fractions, we shall 
get the true value of the expression by considering them all ; 
when their number is infinite, only an approximate value can be 
found. 



CHAV. IX.] CONTINUED FRACTIONS. 279 

220. We will now explain the manner in which any approxi- 
mating fraction may he found from those which precede it. 

(1) . . . . = — 1st app. fraction. 

^ ^ a a 

(2) - - ;- - . - = -T-,— r 2d app. fraction. 

^ ' 1 o6 + 1 

(3) - - . - - . , .^ , . . 3d app. fraction. 



a + 



b-\- 



T 
c 

By examining the third approximating fraction, we see that 
its numerator is formed by multiplying the numerator of the 
preceding approximating fraction by the denominator of the 
third integral fraction, and adding to the product the numerator 
of the first approximating fraction : and that the denominator 
is formed by multiplying the denominator of the preceding 
approximating fraction by the denominator of the third integral 
fraction, and adding to the product the denominator of the 
first approximxating fraction. 

Let us now assume that the [n — l)^'^ approximating fraction 
is formed from the two preceding approximating fractions by the 

P O R 

same law, and let — , — , and ^, designate, respectively, the 

{71 — 3), {n — 2), and (n — 1), approximating fractions. 

Then, if m denote the denominator of the (n — l)*^ integral 
fraction, we shall have from the assumed law of formation, 
E Qm+P 



R Q'm-\-P" 



(1). 



1 c» 

Let us now consider another integral fraction — , and suppose — 
to represent the n*^ approximating fraction.' It is plain that 
we shall obtain the value of — , from that of --, by simply 

changing -- iiito --, or, by substituting m-\-— for m, in 

Tit J. 7i 

m \ 

n 

equation (^^ ' 



280 ELEMENTS OF ALGEBRA. [CHAP. IX 



("""-i) 



^ _ ^ \"^ ' ^^ / "^ _ {Qm-\-P)n-\- Q _En-\-Q 
whence, - _ ^^.^^^ _^^,^^^ ^ ^, _ ^,^^_^ ^, 



,(,+i) 



e '^^4--- +^' 



Hence, if the law assumed for the formation of the (w — 1)*'* ap^ 
proxi mating fraction is true, the same law is true for the forma- 
tion of the n^^ approximating fraction. But we have sho"«T3 
that the law is true for the formation of the third ; hence, it 
must be true for the formation of the fourth; heing true for 
tlif' fourth, it is true for the fifth, and so on ; hence, it is gen 
eral. Therefore, 

The numerator of the n*^ ajjproximating fraction is formed hy 
multiplying the numerator of the preceding fraction by the denom 
inaior of the ??*'' integral fraction, and adding to the product the 
numerator of the {ji — 2)*^ approximating fraction ; and the denom- 
inator is formed according to the same law^from the two preceding 
^nominators. 

221. If we take the difference between the first and second 
approximating fractions, we find, 

1 b __ah-\-\ —ab _ +1 

T ~ a6 + 1 ~ a{ab + 1) ~ a{ab + 1) ' 
aud the difference between the second and third is, 
b bc4-\ - 1 



ab -\- \ {ab -f l)c -\- a {ab + \) [{ab 4 l)c + a]* 
In both these cases we see that the difference between two 
consecutive approximating fractions is numerically equal to 1, 
divided by the product of the denominators of the two fractions 
To show that this law is general, let 

P_ Q_ R_ 

F'' Q'' B^' 

be any three consecutive approximating fracUons. Then 

, q R R'q-pq' 



CHAP. IX. J CONTINUED FKACTIONS. 281 

But E = Qm -f- P, and B' — Q'm -j- P' (Art. 220). 
Substituting these values in the .ast equation, we have, 

q' R~ Rq' ' 

or, reducing, 

q' R' R'q' * 

Now, if {Pq' — P/q) is equal to ± 1, then (P' q — Pq') must 
be equal to qp 1 ; that is, 

If the difference between the [n — 2) and the {n — 1) fractions^ 
is formed by the assumed laiv, then the difference between the 
(n — 1)''* and the n*^ frictions must be formed by the same law. 

But we have shown that the law holds true for the difference 
between the second and third fractions ; hence, it must be true for 
the difference between the third and fourth; being true for the 
difference between the third and fourth, it must be true for the 
difference between the fourth and fifth, and so on ; hence, it is 
general : that is, 

The difference between any two consecutive approximating frac- 
tions, is equal to ±1, divided by the product of their denom- 
inators. 

When an approximating fraction of an even order v^ taken 
from one of an odd order, the upper sign is used : when one 
of an odd order is taken from .one of an even order, the 
lower sign is used. 

This ought to be the case, since we have shown that everj 
approximating fraction of an odd order is greater than the true 
value of the continued fraction, whilst every one of an even 
order is less. 

222. It has already been shown (Art. 218), that each of the 
approximating fi^actions of an odd order, exceeds the true value 
of the continued fraction ; while each one of an even order 
is less than it. Hence, the difference between any two con- 
Becutive approximating fractions is greater than the difference 



282 ELEMENTS OF ALGEBRA. ♦ [CHAP. IX. 

between either of them and the true value of the contmued 
fraction. Therefore, stopping at the n*^ approximating fraction, 
the result will be the true value of the fraction, to within less 
than 1 divided by the denominator of that fraction, multiplied 
by the denominator of the approximating fraction which follows. 

Thus, if Q^ and R^ are the denominators of consecutive ap- 
proximating fractions, and we stop at the fraction whose de- 
nominator is Q', the result will be true to within less than . 

But, since a, 6, c, c?, (fee, are entire numbers, the denominatoi R' 
will be greater than Q% and we shall have 

— < — • 



— , that is. 



hence, if the result be true to within less than , it will 

certainly be true to within less than the larger quantity 

The approximate result wMch is obtained, is true to within 
less than 1 divided by the square of the denominator of the last 
approximating fraction that is employed. 

829 
347' 



223 • If we take the fraction 7^^^, we have, 



IS = 2+ 1 



^'' 2 + -L 



1 + 



1 + 1 



^*h 



Here, we have in the quotient the whole number 2, which 
may either be set aside, and added to ihe fractional part after 
its value shall have been found, or we may place 1 under it 
^or a denominator, and t^'eat it as an approximating fraction. 



CHAP. IX.J . EXPONENTIAL QUANTITIES. 283 

Solution of the Equation a^ = b. 

224. An equation of the form, 

a* = 6, 
^ called an exponential equation. The object in solving this 
equation is, to find the exponent of the power to which it is 
necessary to raise a given number a, in order to produce 
another given number h. ^ 

225. Suppose it were required to solve the equation, 

2 = 64. ff 

By raising 2 to its different powers, we find that 

6 

2 = 64 ; hence, x = 6 

will satisfy the equation. ^ 

Again, let there be the equation, 

3 = 243, in which x = b. 

Now, so long as the second member 6 is a perfect power of 
the given number a, the value of x may be obtained by trial, 
by raising a to its successive powers, commencing at the first, 
(^')r the exponent of the power will be the value of x. 

226. Suppose it were required to solve the equation, 

X 

2=6. 
By making a; = 2, and x = S,' we find, 

2 3 

2=4 and 2=8; 
from which we perceive that the value of x is comprised be- 
tween 2 and 3. 

Make, then, x z=z2 -\ -., in which a/ > 1. 

X 

Substituting this value in the given equation, it becomes, 

2+1 JL 

2 ^' r= 6, or 22 X 2^^ = 6 ; hence, 

4 - 2 ' 





284 ELEMENTS OF ALGEBRA. • [CHAP. IX 

and by changing the order of the members, and raising both i 
to the a/ power, 



iw- - 



To determine x\ make a/ successively equal to 1 and 2; we 
find, 

therefore, a/ is comprised between 1 and 2. 
Make, a/ = 1 -|- — , in which a;^^ > 1. 

By substituting this value in the equation j — 1 =2, 

we find, (1)'^^ - 2; h^^^^^' Y X (1)^ = ^» 

and consequently, I— I = -^. 

4 3 

The supposition, a/^ = 1, gives -^ < ttJ 

16 3 

and a/' = 2, gives ■g->-2> 

therefore, a/^ is comprised between 1 and 2. 

Let x^^ = 1 -{- —jjj ; then, 

ar 

whence, ^-j = -. 

If we make xf'' = 2, we have 

l¥/ ~ 64 ^ 3 ' 

and if we make a/^^ = 3, we have 



/9^\3_729 4^ 
\8/ ~5T2^¥' 



CHAP. IX.J ' EXPONENTIAL QUANTITIES. 285 

therefore, xf^' is comprised between 2 and 3. 
Make xf^' = 2 H , and we have 

and consequently, ^— j = — . 

Operating upon this exponential equation in the same manner 
as upon the preceding equations, we shall find tM^o entire num 
bers, 2 and 3, between which x^"^ will be comprised. 

Making 

x" can be determined in the same manner as a;^^, and so on. 
Making the necessary substitutions in the equations 

we obtain the value of x under the form of a whole number, 
plus a continued fraction, 

1 



a; = 2 + 






2 + 1 



hence, we find the first three approximating fractions to be 

111 
1' 2' 5' 

and the fourth is equal to 

5x3 + 2-12 ^^'^'- ^^^^^ ^ 

which is tne true value of the fractional part of x to within 
less than 

(4' '^ ll4(^«-2^^>- 



286 ELEMENTS OF ALGEBRA. [CHAP. DT, 

Therefore, 

7 31 " 1 

x = 2+—=—- = 2.58333 + to within less tlian — -, 
12 12 144 

and if a greater degree of exactness is required, we must take 
a greater number of integral fracticns. 

EXAMPLES, 

3 = 15 - - a; = 2.46 to within less than 0.01. 

(10)* =3 - - - a; = 0.477 " « 0.001. 

2_ 
3 



5' =~ , . . xz= — 0.25 " " 0.01. 



Of Logarithms. 

227. If we suppose a to preserve a constant value in the 
equation 

whilst a is made, in succession, equal to every possible num- 
ber, it is plain that x will undergo changes corresponding to 
those made in N. By the method explained in the last arti- 
cle, we can determine, for each value of JV, the corresponding 
value of X, either exactly or approximatively. 

The value of x, corresponding to any assumed value of the 
number iV, is called the logarithm of that number ; and a is 
called the base of the system in which the logarithm is taken. 
Hence, 

The logarithm of a number is the exponent of the power to which 
it is necessary to raise the base, in order to produce the given number. 
The logarithms of all numbers corresponding to a given base constitute 
a system of logarithms. 

Any positive number except 1 may be taken as the base 
of a system of logarithms, and if for that particular base, we 
suppose the logarithms of all numbers to be computed, they 
will constitute what is called a system of logarithms. Hence, 
we see that there is an infinite number of systems of loga- 
rithms. 



CHAP. IX.] THEORY OF LOGARIl HMS. 287 

228. The base of the common system of logarithms is 10, 
and if we designate the logarithm of any number taken in 
that system by log, we shall have, 



(10)0 = 1 

(10)1 = 10 

(10)2 ^ 100 

(10)3= 1000 



whence, log 1=0 
whence, log 10 = 1 
whence, log 100 = 2 
whence, log 1000 = 3 
&c., • &c. 

We see, that in the common system, the logarithm of any 
number between- 1 and 10, is found between and 1. The 
logarithm of any number between 10 and 100, is between 1 and 
2 ; the logarithm of any number between 100 and 1000, is be- 
tween 2 and 3 ; and so on. 

The logarithm of any number, which is not a perfect power 
of the base, will be equal to a whole number, plus a fraction, 
the value of which is generally expressed decimally. The entire 
part is called the characteristic^ and sometimes the index. 

By examining the several powers of 10, we see, tl^at if a 
number is expressed by a single figure, the characteristic of its 
logaT'ithm Avill be ; if it is expressed by two figures, the 
characteristic of its logarithm will be 1 ; if it is expressed by 
thre^ figures, the characteristic will be 2 ; and if it is expressed 
by n places of figures, the characteristic will be n — \. 

If the number is less than 1, its logarithm will be negative, 
awd by considering the powers of 10, which are denoted by 
negative exponents, we shall have, 

(10)-= ^ =.i; 

^'<'= 1^0 =-''-^ 
(10)"^ = _i_ ^.001 

^ ^ 1000 

&c., &c. 

Here we see that the logarithm of every number between 1 and 
,1 will be found between and — 1 ; that is, it will OQ equal to 
— 1, plus a fraction less than 1. The logji.rithm of any number 



whence, 


log 


.1 = 


-1. 


whence, 


log 


.01 = 


— 2. 


whence, 


log 


.001 = 


-3. 




&c 


, &c. 





288 



ELEMENTS OF ALGEBRA. 



[CHAP. IX. 



between .1 and .01 will be between — 1 and —2; that is, it 
will be equal .to — 2, plus a fraction. The logarithm of any 
number between .01 and .001, will be between — 2 and — 3, 
or will be equal to — 3, plus a fraction, and so on. 

In the first case, the characteristic is — 1, in the second — 2, 
in the third — 3, and in general, the characteristic of the logarithm 
of a decimal fraction is negative^ and nnmericalhj 1 greater than 
the number of O's uhich immediately follow the decimal jjoint. The 
decimal part is always positive, and to indicate that the negative 
sign extends only to the characteristic, it is generally written 
over it : thus, 
log 0.012 = 2.079181, which is equivalent to — 2 + .079181. 

228* t A table of logarithms, is a table containing a set of 
numbtrs, and their logarithms so arranged that we may, by its 
aid, find the logarithm of any number from 1 to a given num- 
ber, generally 10,000. 

The following table shows the logarithms of the numbers, from 
I to 100. 





' '--' 


N. 
26 


Lojr. 


X. 

51 


Ldg. 


76 


I,..-. 


o.oouooo 


1.414973 


1.707570 


1.880814 


2 


301030 


27 


1.431364 


52 


1716003 


77 


1886491 


3 


477121 


28 


1.447158 


53 


1.724276 


78 


1.892U95 


4 


0.6021)60 


29 


1.462398 


54 


1.732394 


79 


1.897627 


5 


69S970 


30 


1.477121 


55 


1.740363 


80 


1.903090 


~6 


0.77815,1 


31 


1.491362 


56 


1.748188 


sT 


r908485 


n 


0.845098 


32 


1.505150 


57 


1.755875 


82 


1913814 


8 


903090 


33 


1.518514 


58 


1.763428 


83 


1919078 


9 


0.954243 


34 


1.531479 


59 


1.770852 


84 


1.924279 


10 


1000000 . 


35 


1.544068 


60 


1.778151 


85 


1.929419 


11 


1.041393 


36 


1.556303 


61 


1.785330 


86 


1.934198 


12 


1.079181 


37 


1.568202 


62 


1.792392 


87 


1.939519 


13 


1.113943 


38 


1.579784 


63 


1.799341 


88 


1.944483 


J4 


1.146128 


39 


1.591065 


64 


1.806180 


89 


1 949890 


15 


1.176091 


40 


1,602060 


65 


1.812913 


90 


1.954243 


16 


1204120 


il 


1612784 


66 


1819544 


91 


1 959041 


17 


1.230449 


42 


1.623249 


67 


1.826075 


92 


1.963788 


18 


1.255273 


43 


1.633468 


68 


1.832509 


93 


1968483 


19 


1 278754 


44 


1 613453 


69 


1838849 


94 


1.973128 


20 
21 


1.30 i 030 
1322219 


45 
46 


1.65:^213 
1.662758 


70 
71 


1.845098 
1851258 


95 
96 


1.977'-24 


1.982271 


22 


1.342423 


47 


1.672098 


72 


1.857333 


97 


1986772 


23 


1.361728 


48 


1.681241 


73 


1.863323 


98 


1991226 


24 


1380211 1 


49 


1.690196 


74 


1.869232 


99 


1 995635 


25 


1.397940 ! 


50 


1698970 


75 


1.8X5061 


100 


2.0(10000 



CHAP. IX.] THEORY OF LOGARITHMS. 289 

When the number exceeds 100, the characteristic of its loga- 
rithm is not written in the table, but is always known, since 
it is 1 less than the number of places of figures of the given 
number. Thus, in searching for the logarithm of 2970, in a table 
of logarithms, we should find opposite 2970, the decimal part 
,472756. But since the number is expressed by four figures, 
the characteristic of the logarithm is 3. Hence, 

log 2970 = 3.472756, 
and by the definition of a logarithm, the equation 
a* =z iV, gives 

^ 103.472756 -_ 2970. 

General Projperties of Logarithms. 

229. The general properties of logarithms are entirely inde- 
pendent of the value of the base of the system in which they 
are taken. In order to deduce these properties, let us resume 

the equation, 

a' = N, 

in which we may suppose a to have any positive value ex- 
cept 1. 

230. If, now, we denote, any two numbers by N' and N"^ 
and their logarithms, taken in the system whose base is a, 
by a;' and x^\ we shall have, from the definition of a logarithm, 

a^' z=N' ..... (1), 

I and, a'" — N'' (2). 

If we multiply equations (1) and (2) tcgether, member liy 
member, we get, 

ox'-t-r" ^ ^y/ X ^r^^ . . . (8). 

But since a is the base of the system, we have from the 

definition, 

of -\-x" = log {N' X N'') ; that is, 

The logarithm of the product of two numbers is equal to Hie 

sum of their logarithms^ 

19 



290 ELEMENTS OF ALGEBRA. [CHAP. tX 

231. If we divide equation (!) by equation (2), member by 
member, we have, 

N' 

But, from the definition. 



a/ — a/' = log 



The logarithm of th^ quotient which arises from dividing one 
number hy another is equal to the logarithm of the dividend mimis 
the logarithm of the divisor. 

232. If we raise both members of equation (1) to the w** 
power, we have, 

a^'' = W (5). 

But from the definition, we have, 

^ nx' = log (iV'") ; that is, 

The logarithm of any power of a number is equal to the 
logarithm of the number multiplied by the exponent of the power. 

233. If we extract the n^^ root of both members of equation 
(1), we shall have, 

x' 1 

a" z=z{Ny= ^5^ - - (6). 
But from the definition, 

^ = log(V^); that is, 

The logarithm of any root of a number is equal to the loga- 
rithm of the number divided by the index of the root. 

234* From the principles demonstrated in the four preceding 
articles, we deduce the following practical rules : — 

First, To multiply quantities by means of their logarithms. 

Find from a table, the logarithms of the given factors, fake 
the sum of these logarithms, and look in the table for the cor- 
.respmiding number ; 'his will be the product required. 



log 21 - 




- 1.322219 


log 4 - 


21 


- 0.602060 



CHAP. IX.j THEORY OF LOGAEITHMS. 291 

Thus, log 7 0.845098 

log 8 0.903090 

log 56 1.748188 ; 

hence, 7 X 8 = 56. 

Second. To divide quantities by means of their logarithms. 

Find the logarithm of the dividend and the logarithm of the 
divisor, from a table ; subtract the latter from the former, and 
look for the number corresponding to this difference ; this will be 
the quotient required. 

Thus, log 84 - 1.924279 



. hence, 



Third, To raise a number to any power. 

Find from a table the logarithm of the number, and multiply it 
by the exponent of the required power ; find the number corres- 
pondirig to this product, and it will be the required power. 

Thus, log 4 0.602060 

3 

log 64 - - - ... 1.806180 ; 
hence, (4)^ := 64. 

Fourth, To extract any root of a number. 

Find from a table the logarithm of the number, and divide 
this by the index of the root ; find the number corresponding to 
this quotient, and it will be the root required. 

Th'is, log 64 1.806180(6 

log 2 .301030; 

hence, l/^ = 2. 

By the aid of these principles, we may write the foilowinff 
equivalent expressions : — 



Log 


m 


Log 


(«•" .b'^.cP . 


Log 


(a2 _ a:2) 



292 ELEMENTS OF ALGEBRA, [CHAP. IX. 

Log (a .b .t .d . . . .) = log a -f log b -\- log c 

= log a 4- log b -{- log c — log d — log & 

. ) = 7?^ log a + ?2 log 5 + /) log c + . . . . 
= log (a -f a:) + log {a — x). 
Log y (a2 - P) — 1. log (a + re) + I- log (a — a;>. 

Log (a3 X t/^) = 3f log a. 

234, We have already explained the method of deteruiining 
the characteristic of the logarithm of a decimal fraction, in the 
common system, and by the aid of the principle demonstrated 
m Art. 231, we can show 

That the decimal part of the logarithm is the same as the decimal 
part of the logarithm of the mnnerator^ regarded as a whole number. 

For, let a denote the numerator of the decimal fraction, and 
let m denote the number of decimal places in the fraction, then 
will the fraction be equal to 

a 

Aid its logarithm may be expressed as follows : 

log —^ — log^ a — log (10)'" = log a — m log 10 =: log a — m, 

but m is a whole number, hence the decimal part of the loga 
rithm of the given fraction is equal to the decimal part of 
log o, or of the logarithm of the numerator of the gi/eu 
fraction. 

Hence, to find the logarithm of a decimal fraction from the 
Gcmimon table, 

Look for the logarithm of the number^ neglecting the decimal 
point, and then prefix to the decimal part found a negative cliarac- 
teristic equal to 1 mo)'e than the number of zeros which immediately 
follow the decimal point in the given decimal. 

The rules given for finding the characteristic of the logarithms 
talven in the common system, will not apply in any other 
sj-stem, nor could we find the logarithm of decimal fractions 



CHAP. IX.: THEORY OF LOGARITHMS. 298 

directly from the tables in any other system than that Mhose base 
is 10. 

These are some of the advantages which the common system 
possesses over every other system. ., 

235. Let us again resume the equation 

«* 1= jsr. 

1st. If we make iV"= 1, a; must be equal tc 0, since a^ = 1 ; 
that is, 

The logarithm of 1 in any system is 0. 

2d. If we make N =z a, x must be equal to 1, since a^ — a- 
that is, 

Whatever he the base of a system^ its logarithm^ taken in tfvat 
system, is equal to 1. 

Let us, in the equation, 

a* = iV, 

First, suppose a > 1. 

Then, when N = 1, a; = 0; when iV> 1, a; > ; when iV^< i; 
r < 0, or negative ; that is, 

In any system tuhose base is greater than 1, the logarithms of 
all numbers greater than 1 are positive, those of all members less 
than 1 are negative. 

If we consider the case in which i\"< 1, we shall have 
f^-x — jsr or — = iV. 

N#w, if iV diminishes, the corresponding values of x must 
increase, and when iV becomes less than any assignable quaii- 
tity, or 0, the value of x must be oo : that is. 

The logarithm of 0, in a system whose base is greater than I, 
is equal to — oo. 

Second, suppose a < 1. 

Then, when N=l, x = 0], when iV< I, a: > ; w^heniV>l, 
X < 0, or negative : that is, 



294 ELEMENTS OF ALGEBRA. [CHAP. IX 

In any system whose hose is less than 1, the logarithms of all 
numhers greater than 1 are negative, and those of all numbers less 
than 1 oj'e positive. 

* If we consider the case in which iV< 1, we shall have a* = iV, 
in which, if N be diminished, the value of x must be increased ; 
wid finally, when iV^ m 0, we shall have a: = go : that is, 

The logarithm of 0, in a system whose base is less than 1, is 
tqual to + 00. 

Finally, whatever values we give to a?, the value of a' or 
N will always be positive; whence we conclude that negative 
numhers have no logarithms. 

Logarithmic Series. 
236 1 The method of resolving the equation, 

explained in Art. 226, gives an idea of the construction of loga- 
rithmic tables ; but this method is laborious when it is necessary 
to approximate very near the value of x. Analysts have dis- 
covered much more^ expeditious methods for constructmg new 
tables, or for verifying those already calculated. These methods 
consist in the development of logarithms mto series. 
If we take the equation, 

a^ = y, 
and regard a as J;he base of a system of logarithms, we shall 

have, 

logy = x. 

The logarithm of y will depend upon the value of y, and 
also upon a^ the base of the system in which the logarithms 
are taken. 

Let it be required to develop log y into a seiies arranged 
according to the ascending powers of y, with co-efficients that 
are independent of y and dependent upon a, the base of the 
By stem. 



CHAP. IX.J LOGARITHMIC SERIES. 295 

Let US first assume a development of the reqiiired form, 
log y z:^ if + Ny + P3/2 + (>y3 4. &c., 

in which M^ iV, P, &c. are independent of y, and dependent 
upon a. It is now required to find such values for these co- 
efficients as will make the development true for every »-alue 
of y. 

Now, if we make y = 0, log y becomes infinite, and is either 
negative or positive, according as the base a is greater or less 
than 1, (Arts. 234 and 235). But the second" member under 
this supposition, reduces to M^ a finite number : hence, the 
development cannot be made under that form. 

Again, assume, 

log y^ My -{- Ny"^ + Py^ + &c. 

a we make y = 0, we have 

log =1 that is, ± 00 nr 0, 

which is absurd, and therefore the development cannot be made 
under the last form. Hence, we conclude that, 

The logarithm of a number cannot be developed according to 

the asxending poivers of that number. 

Let us write (1 -|- y), for y in the first member of the 
assumed development; we shall have, 

log (1 4- y) .= i/y + Ny'^ + Py3 + Qy^ + &C. - - (1), 

making y = 0, the equation is reduced to log 1 =: 0, w^hich does 
tiot present any absurdity. 

Since equation (1) is true for any value of y, we may write 
z for y; whence, 

log (1 -f 2;) = i/0 + Nz^ + Pz^ + ^2* + &c. - - - (2). 

Subtracting equation (2) from equation (1), member from mem- 
oer, we obtain, 

log (1 + y) - log (1 + ^) = M{y - e) -f iV^(y2 _ ^2) .^ p(^3 „ ^) , 

+ Q(y' - ^0 - - ■ (3). 



296 



ELEMENTS OF ALGEBRA. 



[CHAP. DC 



The second member of this equation is divisible by (y — z)^ 
let us endeavor- to place the first member under such a form 
that it shall also be divisible by [y — z). We have, 

log (1 + y) - log (1 + ^) = log {L±^) = log (l + 1^). 



But since 



1 + 



can be regarded as a single quantity, we may 



substitute it for y in equation (1), which gives. 

Substituting this development for its equal, in the first member 
of equation (3), and dividing both members of the resulting 
equation by (y — z), and we have, 



M 



(ri-.) 



+ iV^ 



{y 



0_,pO/ 



-\- &cG. = M -\- N{y + b) 



{i + zf (1+^)3 

+ P(y2 + 2/2 + £■2) 4- &c. 

Since this equation is true for all values of y and z^ make 
ar =y, and there will result 

M 



\ + y 



Jf + 2Ny 4- 3P2/2 + 4$y3 _j_ 57^^/* + &c. 



Clearing of fractions, and transposing, we obtain. 



-Jf-f M 



y+ 3P 

-f 2N 



2/^ + 4^ 
+ 3P 



2/3 + 5P 
+ 4^ 



2/*+ &c.=0, 



and since this equation is identical, we have, 
M — Jif = ; whence, M = M\ 

M 



2iV+ M=^\ whence, iV^=: — 



2 ' 



3P+2iV^=0; whence, P=-^=|:; 



4$-f3P = 0; whence, 



3P 
4 
&;c. 



4* 



CHAP. IX.-j LOGARITHMIC SERIES. 297 

The^law of the co-efficients in the development is evident; 

M 
the co-efficient of ?/" is ^ — , according as n is even or odd. 

Substituting these values for iV, P, Q^ &c., in equation (1), 
we find for the development of log (1 -\- y)\ 

M M M 
, 'log(H-y) = My - —y'^ ■\- -y"^ - -y'' . . &c. 

= 4-^^^ ?..-.) --(4), 

which is called the logarithmic series. 

Hence, we see that the logarithm of a number may. be 
developed into a series, according to the ascending powers of 
U number less than it by 1. 

In the above development, the co-efficients have all been de- 
termined in terms of M. This should be so, since if depends 
upon the base of the system, and to the base any value may be 
assigned. By examining equation (4), we see that, 

The expression for the logarithm of any number is composed of 
two factors, one dependent on the munher, and the other on the 
base of the sy stein in which the logarithm is taken. 

The factor which depends on the base, is called the modulus 
of the system of logarithms. 

237* If we take the logarithm of I -\- y in a new system, 
and denote it by Z(l + y), we shall have, 

?(l + y)=Jf'(y-|+^-^ + ^-&c.) - - (5), 

in which M' is the modulus of the new system. 

Jf we suppose y to have the same value in equations (4) and (5), 
and divide the former by the latter, member by member, we have 

I {I + y) : log {\ -\- y) : : M' : M ; hence, 
The logarithms of the same number, taken in two different systems, 
are to each other as the moduli of those syst^?ms. 



298 ELEMENTS OF ALGEBRA. [CHAP. IX. 

/ 
238* Having shown that the modulus and base of a system 

of logarithms are mutually dependent on each other, it follows, 

that if a value be assigned to one of them, the corresponding 

7alue of the other must be determined from it. 

If then, we make the modulus 

M' = 1, 
the base of the system will assume a fixed value. The system 
of logarithms resulting from such a modulus, and such a base, is 
called the Naperian System. This was the first system known, 
and was invented, by Baron Napier, a Scotch mathematician. 

If we designate the Naperian logarithm by Z, and the loga- 
rithm in any other system by log, the above proportion becomes, 

l{l-\-y) : log(l+y) : : 1 : if ; 
whence, M x l{l -{- y) — log (1 + y). 

Hence, we see that, 

The Naperian logarithm of any number, multiplied hy the modu- 
lus of any other system, will give the logarithm of the same number 
in that system. 

The modulus of the Naperian System being 1, it is found most 
convenient to compare all other systems with the Naperian ; and 
hence, the modulus of any system of logarithms, is 

The number by which if the Naperian logarithm of any 
number be multiplied^ the product will be the logarithm of the 
same number in that system. 

239. Again, M x I {\ -\- y) = log {I -\- y), gives 

^(1+,) = 1M(H^); that is, 

The logarithm of any number divided by the modulus of its 
system, is equal to the Naperian logarithm of the same number. 

240. If we take the Naperian logarithm and make y = 1 
ea nation (5) becomes, 

,2 = i-i- + l_l+i-... 



CHAP. IX.] LOGAKITHMIC SERIES. 299 

A series Trhich does not converge rapidly, and in whicli it would 
be necessary to take a great number of terms to obtain a near 
approximation. In general, this series will not serve for deter- 
mining the logarithms of entire numbers, since for every number 
greater than 2 we should obtain a series in which the terms 
would go on increasing continually. 

241* In order to deduce a logarithmic series sufficiently con 
verging to be of use in computing the Naperian logarithms 
of numbers, let us take the logarithmic series and make 
M'=l. Designating, as before, the Naperian logarithm by l^ we 
shall have, ^ 

;(i+y) = y-| + |-|+|_&c. .-. (1). 

If now, we write in equation (1), —7/ for ?/, it becomes, 

Subtracting equation (2) from (1), member from member, 
we have, 

?(l+y)-^(l-y) = 2(2/ + ^+|^-» y +y + &c.)-. (3). 

But, 
1(1 -{.y)-l{l^y) = l (^^) ; whence, 

ye 1 1 + 2/ ^ 4- 1 , -,, , 

It now we make = , we shall have, 

1 — y ' ' 

(1 + y)s = (1 — y) (z + 1), whence. 



2z -h 1 

Substituting these values in equatior (4), and observing that 
l{^-^) = K^ + 1) - '^, ^e find, 



300 ELEMENTS OF ALGEBRA. [CHAP. IX 

;(. + l)-L,^2(^-^ + 3^3+g^,+ &c.)(5), 
or, by transpositioii, 

Let us make use of formula (6) to explain the method of 
computing a table of Naperian logarithms. It may be remarked, 
that it is only necessary to compute from the formula the 
logarithms of prime numbers; those of other numbers may be 
found by taking the sum of the logarithms of their factors. 

The logarithm of 1 is 0. If now we make = 1, we can 
find the logarithm of 2 ; and by means of this, if we make 
z ■= 2, we can find the logarithm of 3, and so on, as exhibited 
below. 

Zl =0 - .... -^ -.-.... . =0.000000; 

^^-Ki + ai + ^ + T^-") " ■ • =^-^^^14^' 

Z3 = 0.693147 + 2 (A + 3^^ + ^ + ^ . . . )= 1.098612 

Z4 = 2xZ2 =1.386294 

lf> = 1.386294 + 2(1 + ^+^ + ^...)= 1.609437 

/6 = Z2+Z3 .= 1.791759 

n = 1.79m9 + 2(l + g--1^3+^,+ ...)=1.945910 

/8 = /4-hZ2 - - =2.079441 

/9 = 2x/3 - - - . =2.197224 

110=15+12 =2.302585 

&c. &c. 

in like manner, we may compute the Naperian logarithms 
of all numbers. Other formulas may be deduced, which are 



CHAP. IX.] LOGARITHMIC SERIES. 301 

more rapidly joaverging than the one above given, but this 
serves to show the facility with which logarithms may be com- 
puted. 

241*. We have already observed, that the base of the common 
system of logarithms is 10. We will now find its modulus. 
We have, 

/(I + y) : log (1 + y) : . 1 : 3f (Art. 238). 

If we make y = 9, we shall have, 

110: log 10 : : 1 : M. 

But the no = 2.302585093, and log 10= 1 (Art. 228); 

hence, M = = 0.434294482 = the modulus of the 

common system. 

If now, we multiply the Naperian logarithms before found, by 
this modulus, we shall obtain a table of common logarithms 
(Art. 238). 

All that now remains to be done, is to find the base of the 
Naperian system. If we designate that base by e, we shall have 
(Art. 237), 

le : loge : : 1 : 0.434294482. 

But le = l (Art. 235): hence, 

1 : loge : : 1 : 0.434294482; 

hence, log e = 0.434294482. 

But as we have already explained the method of calculating 
the common tables, we may use them to find the number whose 
logarithm is 0.434294482, which we shall find to be 2.718281828 ; 
hence, 

6 = 2.718281828 

We see frcm the last equation but one, that 

The modulus of the common system is equal to the common loga 
rithm of the Naperian base. 



802 ELEMENTS OF ALGEBRA. [CHAP. IX 

Of Interpolation, 

242. When the law of a series is given, and several tering 
taken at equal distances are known, we may; by means of 
the formula, 

r^. + „,. + !i^-J),,,- "(;-y-|) ,,+L (1), 

already deduced, (Art. 209), introduce other terms between 
them, which terms shall conform to the law of the series 
This operation is called interpolation. 

In most cases, the law of the series is not given, but only 
numerical values of certain terms of the series, taken at fixed 
intervals ; in this case we can only approximate to the law 
of the series, or to the value of any intermediate term, by 
the aid of formula (1). 

To illustrate the use of formula (1) in interpolating a term 
in a tabulated series of numbers, let us suppose that we have 
the logarithms of 12, 13, 14, 15, and that it is required to find 
the logarithm of 12|-. Forming the orders of ' differences from 
the logarithms of 12, 13, 14 and 15 respectively, and taking 
the first terms of each, 

12 13 14 15 

1.079181, 1.113943, 1.146128, 1.176091, 

0.034762, 0.032185, 0.029963, 

- 0.002577, - 0.002222, 

+ 0.000355, 
we find d, = 0.034762, d^=- 0.002577, d^ = 0.000355. 

If we consider log 12 as the first term, we have also 

a= 1.079181 and n = —. 

Making these several substitutions in the formula, and ne- 
glecting the terms after the fourth, since they are inappieciable. 
we find, 

, T = a + — d,~-—d,-^—d, = log 12i; 



CHAP. IX. J FORMULAS FOR INTEREST. 803 

or, by substituting for c?,, d^^ &;c., their values, and for a its 
value, 

(/..-- ^. . - . 1.079181 

\d^ 0.017381 

ic?2 - - - - - - 0.000322 

Jg-(^3 - - - '- . - - 0.000022 

Log 12} ... . 1.096906 

Had it been required to find the logarithm of 12.39, we 
should have made • n = .39, and the process would have been 
the same as above. In like manner we may interpolate terms 
between the tabulated terms of any mathematical table. 



INTEREST. 

243 • The solution of all problems relating to interest, may 
be greatly simplified by employing algebraic formulas. 

In treating of this subject, we shall employ the following 
notation : 

Let p denote the amount bearing interest, called the principal; 
r " the part of $1, which expresses its interest for 

one year, called the rate per cent.; 
t " the time, in years, that p draws interest ; 
e " the interest of p dollars for t years ; 
'^ " V i^ + the interest which accrues in the time t. 

This sum is called the amount. 

Simple Interest. 

To find the interest of a sum p for t years^ at the rate r, and 
the amount then due. 

Since r denotes the part of a dollar which expresses its in- 
terest for a single year, the interest of p dollars for the same 



304 ELEMENTS OF ALGEBKA. [CHAP. IX. 

lime will be expressed hy pr; and for t years it will be i times 
as much : hence, 

i=ptr - - . ^ ... (1); 

and for the amount due, 

S = p +ptr =p (1^+ tr) - - (2). 

EXAMPLES. 

1. What is the interest, and what the amount of $365 for three 
jears and a half, at the rate of 4 per cent, per annum. Here, 

p = $365 ; 

'■ = 4 = 0.04; 

i =: 3.5 ; 

i = ptr — 3G5 X 3.5 X 0.04 — $51,10 : 
hence, ^ = 365 + 51,10 = $416,10. 

Present Value and Discount at Simple Interest. 

The present value of any sum aS', due t years hence, is the prin- 
cipal p^ which put at Interest for the time t, will produce the 

amount S. 

The discount on any sum due t years hence, is the difference 
between that sum and the present value. 

To find the present value of a sum of dollars denoted by S, due 
t years hence^ at simple interest^ at the rate r; also, the discount. 

We have, from formula (2), 

S=p-\-ptr', 

and since p is the principal which in t years will produce the 
Bum >S, we nave 



CHAP. IX.J FORMULAS FOR INTEREST. 305 

and for the discount, which we will denote by D^ we have 

1. Required the discount on $100, due 3 months honce, at the 
rate of 5^ per cent, per annum. 

S = $100 = IIOO,**" 

t = 3 months = 0.25. 

Hence, the present value p is 

hence, I) =S-'p =100- 98,643 = $1,357. 

Compound Interest 

Compound interest is when the interest on a sum of money- 
becoming dae, and not paid, is added to the principal, and 
the interest then calculated on this amount as on a new 
principal 

To jind the amount of a sum p placed at interest for t yearSy 
compound interest being allowed annually at the rate r. 

At the end of one year the amount will be, 

S =z p -{- pr =1 p(l + r). 

Since compound interest is allowed, this sum now becomes 
the principal, and hence, at the end of the second year, the 
amount will be, 

S' =:;j(l + r) +pr(l + r) = p{l -f r)^. 

Regard p(l -\- rY as a new principal ; we have, at the end 
of the third year, 

S'' =p{l-i-ry-{-pr(l-{-rY=:p{l + rY', 
20 



ELEMENTS OF ALGEBRA. [CHAP. tX, 

flud at the end of t years, 

S = p{\-^tY .... (5). 
Aiid from Articles 230 and 232, we have, 

log S = logp + t log (1 -L r) ; 

and if any three of the four quantities aS, p, t^ and r, are given^ 
the remaining one can be determined. 

Let it be required to find the time in which a sum p will 
double itself at compound interest, the rate being 4 per cent.. 
per annum. 

We have, from equation (5), 

S =p{\ + r)^ 

But by the conditions of the question, 

S=2p=p{l-\-rY', 

hence, 2 = (1 + r)'. 

log 2 0.301030 



and 



• log (1 + r) 0.017033' 
= 17.673 years, 
= 17 years, 8 months, 2 days. 

To find the Discount. 



The discount being the difference between the sum S and jp, 
we have. 



CHAPTER X. 



GENERAL THEORY OF EQUATION?. 



244. Every equation containing but one unknown q lantity 
which is of the 7n*^ degree, m being any positive whole number, 
may, by transposing all its terms to the first member and divid- 
ing by the co-efficient of x^, be reduced to the form 

^m ^ p-^m-\ _|. Q^in~2 J^ ^ ^ ^ ^ ^ Tx -\- U = Q. 

In this equation P, ^, .... J', U^ are co-efficients in the 
most general sense of the term ; that is, they may be positive '^ 
or negative, entire or fractional, real or imngiiiary. 

The last term U is the co-efficient of .r", and is called the 
absolute term. 

If none of these co-efficients are 0, the equation is said to be 
co7riplete ; if any of them are 0, the equ;ition is said to be 
incomplete. 

In discussing the properties of equations of the rn^^ degree, 
involving but one unknown quantity, we Av\[\ hereafter suppose 
them to have been reduced to the form ju i given. 

245. We have already defined the root of ;h. equation (Art. 77) 
to be any expression^ which^ when subatituietl for the unknowih 
quantity in the equation^ will satisfy it. 

"We have shown that every equation of ihe fii-st degree haa 
one root, that every equation of the second degree has two 
roots; and in general, if the two member- of an equation are 
equal, they must be so for at le-ast son c one value of the 



SOS ELEMENTS OF ALGEBFwA. [CHAP. X. 

imkno-wn quantity, either real or imaginary. Sucti value of the 
unknown quantity is a root of the equation : hence, we infer, that 
every equation, of whatever decree, has at least one root 

We shall now demonstrate some of the principal properties 
ef equations of any degree whatever. 

First Property. 
246 1 In every equation of the form 

if a is a root, the first member is divisible by x — a ; and con 
versely, if the first member is divisible by x — 2b, b. is a root of 
tJie equation. 

Let us apply the rule for the division of the first member 
by X — a, and continue the operation till a remainder is found 
which is independent of x ; that is, which does not contain x. 

Denote this remainder by R and represent the quotient found 
by Q', and we shall have, 

.T-" + P^"^-^ . . , . -\- Tx+ TJ— Q,\x — a)-\-R, 

Now, since by hypothesis, a is a root of the equation, if we 
substitute a for x, the first member of the equation will reduce to 
.zero; the term Q'{x — a) will also reduce to 0, and consequently, 
we shall have 

72 = 0. 

But since R does not contain x, its value will not be affected 
by attributing to x the particular value a : hence, the remainder 
JR is equal to 0, whatever may be the value of x, and conse- 
.quenily, the first member of the equation 

^m ^ p^m-l ^ Q^m-'z , , , , ^ Tx -{^ JJ = 0, 

Is exactly divisible by x — a. 

Conversely, if x — a is an exact divisor of the first member 
of the equation, the quotient Q' will be exact, and we shall have 
R = 0: hence, 

x^+Px"^^ .... ^ Tx-^ 11= Q'{x — ay 



CHAP. X.] THEORY OF EQUATIONS. 30^ 

If now, we suppose x — a^ the second member will reduce to 
zero, consequently, the first will reduce to zero, and hence a will 
be a root of the equation (Art. 245). It is evident, from the 
nature of division, that the quotient Q' will be of the form 
a;m-.: ^ p'^m-2 ^B!x->r TJ' -^, 

247« It follows from what has preceded, that in order to dis- 
cover whether any polynomial is exactly divisible by the biiHV 
mial X — a, it is sufficient to see if the substitution of a for ic 
will reduce the polynomial to zero. 

Conversely, if any polynomial is exactly divisible by x — a^ 
then we knOw, that if the polynomial be placed equal to zero, 
a will be a root of the resulting equation. 

The property which we have demonstrated above, enables ua 
to diminish the degree of an equation by 1 when we kiow 
one of its roots, by a simple division; and if two or ix). »rB 
roots are known, the degree of the equation may be still fur ;iver 
diminished by successive divisions. 

EXAMPLES. 

1. A root of the equation, 

a;4 — 25:^2 -f 60a; — 36 = 0, 
is 3 : what does the equation become when freed of this o//t 2 
a;4 — 25a;2 + 60a; — 36 lb - 3 



:x^- 3ar3 


a;3_|.3;r2_io^-j 12. 


+ 3a;3- 


-25a;2 


3a:3 


- 9a;2 




- 16a;2 + 60.r 




- 16a;2 + 48a; 




12a; - 36 




12a; - 36 




Ans. a;3 + 3.1-2 _ lo.r -^ 12 - k\ 



2. Two roots of the equation, 

x^ — 12a;3 + 48a;2 - 68a; +15 = 0, 
are 3 and 5 : what does the equation become when fre^d ^4 
them 1 Ans. x^—4:X-{-l—Q 



SIO ELEMENTS OF ALGEBRA. [CHAP. X. 

8. A root of the equation, 

fa 1 :' what is the reduced equation'? 

Ans. a;2 — 52r -1- G = 0. 

4. Two roots of the equation, 

4a:* - 14^-3 - 5x^ + Six + 6 = 0, 
are 2 and 3 : find the reduced equation. 

Arts. 4:x'^ + 6x -\- 1 = 0. 

Second Property. 

248* Every equation involving hut one unknown quantity, has 
xs many roots as there are units in the exponent which denotes 
its degree, and no more. 

Let the proposed equation be 

x^ + Px"^-^ -f Qx^-'^ j^ , , , j^ Tx-\- 17=0. 

Since every equation is known to have at least one root 
(Art. 245), if we denote that root by a, the first member will 
be divisible by x — a, and we shall have the equation, 
2* -f Pa:*"-! + . . .^ = {x — a) [x^^-^ + P'x'^-^ -f ...)---- (1). 

But if we place, 

x^-i 4- p^x""-^ -\- . . . —0, 
we obtain a new equation, which has at least one root. 

Denote this root by b, and we have (Art. 246), 

Xm-^ -I- P'x'^-2 -[-...=: (a: — 6) (.r'"-2 -f p/'a-m-S _|_ ^ ^ y 

Substituting the second member, for its value, m equation 
(1), we have, 
^ + Pa:"-! + , . . ={x — a) {x — h) (.r'"-^ + P'' x"^^ 4- . . .) • - (2). 

Reasoning upon the polynomial, 

as upon the preceding polynomial, we have 

jr*-2 _j_ p//a;m-3 -f ... - (a; - c) (.'r'"-3 + P'^^x"^-^ +...). 
emd by substitution, 
««.-i-Pa;«-i.f . . . = (a; — a) [x ~ b) {x - c) {x^-^ + P'''x^ - . - (3). 



CHAP. X.I THEORY OF EQUATIONS. 311 

By continuing this operation, we see that for each binomial 
factor of the first degree with reference to a;, that we separate, 
the degree of the polynomial factor is reduced by 1 ; therefore, 
afler m — 2 binomial factors have been separated, the polynomial 
factor will become of the second degree with reference to x^ 
which can be decomposed into two factors of the first degree 
(Art. 115), of the form x — Jc^ x — /. 

Now, supposing the m — 2 flictors of the first degree to have 
already been indicated, we shall have the identical equation, 

^m _|_ p^,n~. ^ _ —{^x — a) {x — b){x — c).. {x — k) {x — I) = ; 

from which we see, that the Jirst member of the proposed equation 
may he decomposed into m binomial factors of the first degree. 

As there is a root corresponding to each binomial factor of 
the first degree (Art. 246), it follows that the m binomial factors 

of the first degree, .t — a, x — b^ x — c , give the m roots, 

a, 6, c . . ., of the proposed equation. 

But the equation can have no other roots than a, 6, c . . . ^, /. 
for, if it had a root a\ different from a, 6, c ..../, it would 
have a divisor x — a\ different from x — a, x — b, x — c...x — ^ 
which is impossible ; therefore, 

Every equation of the m*'* degree has m roots, and can hcive 
no more. 

249. In equations which arise from the multiplication of equal 
factors, such as 

{x — ay {x — by (x — cy {x — d) = 0, 

the number of roots is apparently less than the number of unita 
in the exponent which deno'tes the degree of the equation. But 
this is not really so ; for the above equation actually has ten 
roots, four of which are equal to a, three to b, two to c, and 
one to d. 

It is evident that no quantity a', different from a, 6, c, d^ 
can verify the equation ; for, if it had a root a'', the first mem» 
ber would be divisible by a; — a% which is impossible. 



312 ELEMENTS OF ALGEBRA. FCHAP. X 

Consequence of the Second Pro'perty, 

250. It has been shown that the first member of every equa- 
tion of the m^'^ degree, has m binomial divisors of the firsJ 
degree, of the form 

X — a, X — ^, X — c, . . . a; — h^ x — I. 

If we multiply these divisors together, two and two^ three and 
three, &c., .we shall obtain as many divisors of the second, 
third, &;c. degree, with reference to x, as we can form different 
combinations of m quantities, taken two and two, three and three, 
&c. Now, the number of these combinations is expressed by 

711 — 1 m— Im — 2 ,, 

'"^'""2 — ' ""'' — 2~ ' — 3~~ ' ' ' ^ ^' 

hence, the proposed equation has 

m — 1 

divisors of the second degree ; 

m — 1 m — 2 



2 3 

divisors of the third degree ; 

m — 1 m — 2 m 



m . 



2 3 4 

divisors of the fourth degree ; and so on. 

Composition of Equations. 

251, If we resume the identical equation of Art. 248, 
gfij^p^m-i _}- ^^.m-2 _ , _}_ XJ = (x—a) [x--b){x-~c) ... (.r— I)... 
and suppose the multiplications indicated in the second member 
to be performed, we shall have, from the law demonstrated in 
article 135, the following relations : 
P = — a— b— c — ...— k—l, OT — F = a-{-b + c-{' .. -4-^+^, 

Q =z ab -\- ac -\- be ^ ak -\- kl, 

E = — abc — abd —bed ... — ikL or — M =abc -+- «^<^ + . . .+ ikl^ 



U= d- abed .... ikl, or :h U = ahc . . . ikl 



CHAP. X.] COMPOSITION OF EQUATIONS. .'{13 

Tlie double sign has been placed before the product of a, S, c, &o. 
in the last equation, since the product — a x — b X — c . . x —l^ 
will be plus when the degree of the equation is even^ and minus 
when it is odd. 

By considering these relations, we derive the following conclu- 
sions with reference to the values of the co-efficients : 

1st. Tlie co-efficient of the second term^ with its sign changed^ is 
equal to the algebraic sum of the roots of the equation, 

2d. The co-efficient of tlie third term is equal to the sum of the 
different products of the j-oots, taken two in a set. 

3d. The co-efficient of the fourth term, with its sign changed^ «6 
equal to the sum of the different products of the roots, taken three 
in a set, and so on. 

4th. The absolute term, tvith its sign changed when the equation 
IS of an odd degree, is equal to the continued product of all the 
roots of the equation. 

Conseqiiences. 

1. If one of the roots of an equation is 0, there will be 
uo absolute term ; and conversely, if there is no absolute term, 
?ne of the roots must be 0. 

2. If the co-efficient of the second term is 0, the numerical 
sum of the positive roots is equal to that of the negative roots. 

0. Every root will exactly divide the absolute term. 

It will be observed that the properties of equations of the 
second degree, already demonstrated, conform in all respects to 
the princ pies demonstrated in this article. 

EXAMPLES OF THE COMPOSITION OF EQUATIONS. 

1. Find the equation whose roots are 2, 3, 5, and — 6. 

We have, from the principles already established, the equation 
(x - 2) (a; - 3) (a; - 5) (a: + 6) = ; 
whence, by the application of the preceding principles, we obtain 
the equation, 

a;4 _ 4a:3 _ 29^2 -f 156a; — 180 = 0. 



§14 ELEMENTS OF ALGEBRA. LCHAP. X 

2. What is the equation whose roots are 1, 2, and — 3 ? 

Ans. x^ —7x-\- 6 = 0. 

B. What is the equation whose roots are 3, — 4, 2 -f /s", 
and 2 -y^? Ans. x^ - Zx^ — 15a:2 + 4Qx - 12 = 0. 

4. What is the equation whose roots are 3 4-*/5, 3 — V^, 
and — 6] ' Jns. :i;3 — 32a; + 24 =:: 0. 

5. What is the equation whose roots are 1, — 2, 3. — 4, 5, 
and - 6 ? 

Ans. x^ + Sx^ - 41a;* - Slx^ + 400^2 ^ 444^ _ 720 = 0. 



6. What is the equation whose roots are .... 2 + ^ — 1, 
2 — y — 1, and — 3 r Ans. a;^ — a?2 — 7a; + 15 ^ 

Greatest Common Divisor. 

252. The principle of the greatest common divisor is of fre- 
quent application in discussing the nature and properties of 
equations, and before proceeding farther, it is necessary to inves- 
tigate a rule for determining the greatest common divisor of two 
or more polynomials. 

The greatest common divisor of two or more polynomials is 
the greatest algebraic expression, with respect both to co-efficients 
and exponents, that will exactly divide them. 

A polynomial is ^nme, when no other expression except I 
will exactly divide it. 

Two polynomials are prime loitli respect to each other^ when 
they have no common flictor except 1. 

253t Let A iind B designate any two polynomials arranged 
with reference to the same leading letter., and suppose the 
polynomial A to contain the highest exponent of the leading 
letter. Denote the greatest common divisor of A and B by i>, 
and let the quotients found by dividing each polynomial by D 



CHAP. X.] GREATEST COMMON DIVISOK, 315 

be represented by A' and B' respectively. We shall then have 
the equations, 

^ = ^^ and J=^^; 

whence, A — A^ y. D and B — B' y. L, 

Now, D contains all the factors common to A and B. For, 
if it does not, let us suppose that A and B have a common 
factor d which does not enter Z>, and let us designate the quo- 
tients of A' and B\ by this factor, by A' and B''. We shall 
then have, 

' A=zA''.d.D and B=zB'\d.I>; 

or, by division. 

Since A''^ and ^" are entire, both A and B are divisible by 
rf . D, which must be greater than D, either with respect to its 
co-efficients or its exponents ; but this is absurd, since, by 
hypothesis, D is the greatest common divisor of A and B. 
Therefore, D ■ contains all the factors common to A and B. 

Nor can D contain any factor which is not common to A 
and B. For, suppose D to have a factor d^ which is not con 
tained in A and B, and designate the other factor of D by 2>' ; 
we shall have the equations, 

A = A'.d\D' and B = B\ d\ B' ; 

or, dividing both members of these equations by d', 

4t = A\I)' and ^ = B\D'. 
d' a 

Now, the second members of these two equations being en- 
tire, the first members must also be entire ; that is, both A 
and B are divisible by d\ and therefore the supposition that 
d' is not a common factor of A and B is absurd. Hence, 

1st. The greoiest common divisor of two polynomials contains 
all the factors common to the polyno"nials, and does not contain 
any other factors, '' ^" 



316 ELEMENTS OF ALGEBRA. [CHAP. X 

254. If, now, we apply the rule for dividing A by B, and 
continue the process till the greatest exponent of the leading 
letter in the remainder is at least one less than it is in the 
polynomial B, and if we designate the remainder by M, and 
the quotient found, by Q, we shall have, 

A = Bx Q-hB - - - - (1). 

If, as before, we designate the greatest common divisor of 
A and B by J9, and divide both members of the last equation 
by it, we shall have, 

A B ^ R 

Now, the first member of this equation is an entire quantity, 

R 

and so is the first term of the second member ; hence — 

must be entire; which proves that the greatest common divisor 
of A and B also divides R. 

If we designate the greatest common divisor of B and R by 
i)^, and divide both members of equation (1) by it, we shall have, 
A__B R 

Now, since by hypothesis D^ is a common divisor of B and 
i?, both terms of the second member of this equation are 
entire ; hence, the first member must be entire ; which proves 
that the greatest common divisor of B and R, also divides A. 

We see that D'', the greatest common divisor of B and R^ 
cannot be less than D. since D divides both B and R) nor can 
7>, the greatest common divisor of A and B, be less than D\ 
because I)^ divides both A and B ; and since neither can be less 
than the other, they must be equal ; that is, D = D'. Hence, 

2d. The greatest coramon divisor of two polynomials^ is the same 
as that betiveen the second polynomial and their remainder after 
division. 

From the principle demonstrated in Art. 253, we see that wo 
may multiply or divide oiie polynomial by any factor tbat jg 



UHAP. X.1 GREATEST COMMON DIVISOR, 817 

not contained in the other, without affecting their greatest com- 
mon divisor. 

255, From the principles of the two preceding articles, we 
deduce, for finding the greatest common divisor of two poly- 
nomials, the following 

RULE. 

J. Suppress the monomial factors common to all the terms of the 
first polynomial ; do the same with the second polynomial ; and if 
the factors so suppressed have a common divisor^ set it aside^ as 
forming a factor of the common divisor sought. 

II. Prepare the first polynomial in such a manner that its first 
term shall he divisible by the first term of the second polynomial^ 
both being arranged with reference to the same letter : Apply the 
rule for division, and continue the process till the greatest exponent 
of the leading hotter in the remainder is at least one less than it is 
ill the second polynomial. Suppress, in this remainder, all the 
factors that are common to the co-efficients of the different powers 
of the leading letter ; then take this result as a divisor and the 
second polynomial as a dividend, and proceed as before, 

III. Continue the operation until a remainder is obtained which 
will exactly divide the preceding diviaor ; this last refnainder, mul- 
tiplied by the factor set aside, will be the greatest common divisor 
sought ; if no remainder is found which will exactly divide the 
preceding divisor, then the factor set aside is the greatest common 
divisor sought. 

EXAMPLES. 

1. Find the greatest common divisor of the polynomials 
a3 _ ^2^, _|_ 3aj2 _ 3^,3^ ^nd a2 - bab 4- 46^. 

First Operation. Second Operation, 



fl3__a26-{-3a^»2^363 



Aa^b - ab'^ - Zh^ 



la2 — 5a6 + 4^2 



46 



Ist rem. 19a62 — IQi^ 
or, 1962 (a _ ly 

Hence, a — b is the greatest common divisor. 



hab-\-Ab'^\\a 



— 4ab -I- 462 a —4b 



0. 



318 ELEMENTS OF ALGEBRA. [CHAP. X 

We begin by dividing the polynomial of the higliest degree 
by that of the lowest ; the quotient is, as we see in the above 
table, a + 45, and the remainder lOaS^ — 196^. 

But, 19a62_ 19^;3^ 1962(a -6),^ 

Now, the factor 195^5 will divide this remainder without dividing 
a? — bah + 4^*2 . 
hence, th3 factor must be suppressed, and the question is reduced 
to finding the greatest common divisor between 
a^ — 5ab + 46^ and a — b. 

Dividing the first of these two polynomials by the second, there 
is an exact quotient, a — 45 ; hence, a — b is the greatest com- 
mon divisor of the two given polynomials. To verify this, le^ 
each be divided by a — b. 

2. Find the greatest common divisor of the ' polynomials, 
3a5 _ 5a362 + 2a5* and 2a' - ^aW + 5*. 

We first suppress a, which is a fiictor of each term of the 
first polynomial : we then have, 

3a4 _ ^uW + 26* 1 1 2a4 — 2>aW + 6*. 

We now find that the first term of the dividend will not con- 
tain the first term of the divisor. We therefore multiply the 
dividend by 2, which merely introduces into the dividend a 
factor not common to the divisor, and hence does not affect 
the common divisor sought. We then have, 



Qa'^ — 10a2i2_|_4i. 
6a*- 9aW + 2,b- 



-ia^ 



3a2Z-2 + 6* 



— «2^2^ 54 

_ 62 (a2 - 52). 

We find after division, the remainder — a^b"^ -f- 5* which we 

put under the form — 52 {a? — 52). We then suppress — h\ 

and divide, 

2a* — ZaW -j- 5* I 0.2 — 52 

2a* - 2a262 



52 



— a262 -}- 6* 
Hence, a^ — b^ is the greatest common divisor. 



CHAP. X.J 



GREATEST COMMON DIVISOR. 



819 



3. Let it be required to find tlie greatest common divisor 
between the two polynomials, 

— 363 _|_ 3^52 _ ^25 4. ^3^ and 462 _ 5^5 _^ ^i^ 

First Operation. 



— 1263 -f- 12a62 - 4a'^b -\- 4a^ 



1st rem, 

2d rem. 
or, 



3a62— .a^ -f 4a^ 
12a62 — 4a26 + IQa^ 



'M2 _ r^ab + a2 



36, - 



— 196/26+ 19a3 
19a2(-6 4-a). 

Second Operation. 



462 _ 5^5 4. a"^ 



ah 



- h + a 



-46 + a 



0. 



Hence, — 6 + a, or a — 6, is the greatest common divisor 
In the first operation we meet with a difficult}; in dividing the 
two polynomials, because the first term of the dividend is not 
exactly divisible by the first term of the divisor. But if we 
observe that the co-efficient 4, is not a factor of all the terms 
of the polynomial 

42,2 _ r^ah + a\ 

and therefore, by the first principle, that 4 cannot form a part 

of the greatest common divisor, we can, without affecting this 

common divisor, introduce this factor into the dividend. This 

gives, 

— 1263 + 12a62 — 4a26 + 4a3, 

and then the division of the terms is possible. 

Effecting this division, the quotient is — 36, and the rd 

^mainder is, 

— 3a62 — a26 + 4a3. 

As the exponent of 6 in this remainder is still equal to 

that of 6 in the divisor, the division may be continued, by 

multiplying this remainder by 4, in order to render the division 

of the first term possible. This done, the remainder becomes 

— 12a62 — 4^26 + 16a3; 



820 ELEMEIsTS OF ALGEBRA. [CHAP. X 

which, divided by 46 2 — 5a6 -f c^^ gives the quotient — 3a, 
which should be separated from the first by a comma, having 
no connexion with it. The remainder after this division, is 

- 19a26 + 19a3. 

Placing this last remainder under the form lOa^ (— 6 -{- a), 
and suppressing the factor 19a^, as. formmg no part of the com- 
mon divisor, the question is reduced to finding the greatest 
common divisor between 

452 _ 5^5 _|_ (j2 2iTi^ — h -\' a. 

Di\^ding the first of these polynomials by the second, we 
obtain an exact quotient, — \h -\- a : hence, — 6 -f- «? or a — b^ 
is the greatest" common divisor sought. 

256. In the above example, as in all those in which the 
exponent of the leading letter is greater by 1 in the dividend 
than in the divisor, we can abridge the operation by first mul- 
tiplying every term of the dividend by the square of the co 
efticierxt of the first term of the divisor. We can easily see 
that by this means, the first term of the quotient obtained will 
c-ontain the first power of this co-efficient. Multiplying the 
divisor by .the quotient, and making the reductions with the 
dividend thus prepared, the result will still contain the co-efficient 
as a factor, and the division can be continued until a remainder 
is obtained of a lower degree than the divisor, with reference 
to the leading letter. 

Take the same example as before, viz. : 

— 353 + 3a62 _ aH. + a^ and 452 __ 5^^ _|. ^2^ 
and multiply the dividend by 4^ = 16 ; and we have 

First Operation. 

— 4853 -I- 48a52 — IGa^J 4. IGa^ j452 — 5a5 -f- a^ 



12a52 — 4a25 -f IQa^ 



— 125 - 3a 



1st remainder, — 19a25 + I9a^ 

or, 19a2(— 5-l-a). 



CHAP. X.] GKEATEST COMMON DIVISOR. 321 

Second Operation. 
4^2 — 5a6 + a2||— b -\- a 



ab -{- a^ — 46 4- a 



iid remainder, — 0. 

AVhen the exponent of the leading letter in the dividend 
exceeds that of the same letter in the divisor by two, three, 
&c., multiply the dividend by the third, fourth, &c. power of 
the co-efticient of the first term of the divisor. It is easy to 
see the reason of this. 

257. It may be asked if the suppression of the factors, com 
mon to all the terms of one of the remainders, is absolutely 
necessary^ or whether the object is merely to render the opei*a- 
tions more simple. It will easily be perceived that the suppres- 
sion of these factors is necessary ; for, if the factor 19a^ was not 
suppressed in the preceding example, it would be necessary to 
multiply the whole dividend by this factor, in order to render 
its first term divisible by the first term of the divisor ; but, 
then, a factor would be introduced into the dividend which is 
also contained in the divisor ; and, consequently, the required 
^.rreatest conamon divisor would contain the factor lOa^ which 
should form no pa^'t of it. 

258. For another example, let it be required to find the 
greatest common divisor of the two polynomials, 

a* + ^a% + 4a262 _ Qab^ + 26* and ^a^b + 2a62 _ 263, 
or simply of, 

a* + Za^b + 4a262 — Qab"^ -f 26* and 2a2 -\- ab — 62, 

since the factor 26 can be suppressed, being a factor of tlie 
second polynomial and not of the first. 



First Operation. 
8a* -{- 2ia^ + 32^262 _ 4Sab^ + 166* 



-{- 20a^b + 36a262 — 48a63 -\- 166* 



2a2 4- a6 — 62 
4a2 -f- lOab -h 136« 



-f 26a262 - 38a63 + 166* 

ist remainder, — blab^ + 296* 

or, - 63(51a — 296). 



ELEMENTS OF ALGEBRA. I CHAP. X. 



Second Operation. 
Multiply by 2601, the square of 51. 



5202a2 + 2601a6 - 


- 260162 

- 260162 
-316162 


1 5xa- 296 


5202a2 _ 2958a6 


102a -J- 1096 


1st remainder, + 5559a6 
5559a6 





2d remainder, + 56062. 

The exponent of the letter a in the dividend, exceeding that 
of the same letter in the divisor, by two^ the whole dividend 
is multiplied by 2^ =: 8. This done, we perform the division^ 
and obtain for the first remainder, 

- 51a63 + 296*. 

Suppressing — 6^, this remainder becomes 51a — 296 ; and 
the new dividend is 

2a2 + ah — 62. 

Multiplying the dividend by (51)2 =: 2601, \\^QYi effecting the 
division, we obtain for the second remainder + 56062. Now, it 
resuits from the second principle (Art. 254), that the greatest 
common divisor must be a factor of the remainder after each 
division ; therefore it should divide the remainder 56062. g^^ 
this remainder is independent of the leading letter a : hence, if 
the two polynomials have a common divisor, it must be mde- 
pendent of a, and will consequently be found as a factor in the 
«'50-ef!icients of the different powers of this letter, in each of the 
proposed polynomials. But it is evident that the co-efRcients of 
these powers have not a common factor. Hence, the two given 
polynomials are prime with respect to each other. 

259. The rule for finding the greatest common divisor of two 
polynomials, may readily be extended to three or more poly 
Domials. For, having the polynomials A, B, C, J), <fec., if we 
find the greatest common divisor of A and J?, and then the 
:greatest common divisor of this result and C, the divisor so oh 



CHAP. X.] GREATEST COMMON DIVISOR. 323 

tained will evidently be the greatest comnion divisor of J, i?, 
and (7; and the same process may be applied to the remaining 
polynomials. 

260. It often happens, after suppressing the monomial fiictorj* 
common to all the terms of the given polynomials, and arranging 
the remaining polynomials with reference to a particular letter, 
that there are polynomial factors common to the co-efhcients of 
the different powers of the leading letter in one or both poly- 
nomials. In that case we suppress those factors in both, and if 
the suppressed factors have a common divisor, we set it aside, a3 
forming a factor of the common divisor sought, 

EXAMPLE. 

Let it be required to find the greatest common divisor of the 
two polynomials 

a^d? — c^d? — o^c^ + e*, and Aa^d — ^ac^ -f- 2c^ — 4aC(i. 

The second contains a monomial fictor 2. Suppressing it, 
and arranging the polynomials with reference to c/, we have 
(a2 _ c2) (^2 _ ^2^2 _f_ c*, and {2a? — 2ac) d — ac' + c^. 

By considering the co-efficients, a? — c^ and — arc^ -h c*, in the 
first polynomial, it will be seen that —a^c'^ -\- c* can be put under 
the form — 0^(0^ — c^) : hence, a^ — c^ is a common factor of the 
co-efficients in the first polynomial. In like manner, the co-effi- 
cients in the second, ^a^ —2ac and — ac^ + c^, can be reduced 
to 2o(a — c) and — c^{a — c) ; therefore, a — c is a common 
factor of these co-efficients. 

Comparing the t\vo factors o? — c^ and a — c, we see that the 
last will divide the first; hence, it follows that a — c is a com- 
mon factor of the proposed polynomials, and it is therefore a 
factor of the greatest common ai visor. 

Suppressing o? — c^ in the first polynomial, and a — c in the 
second we obtain the two polynomials, 

d'^ — c2 and 2ad — c^^ 



^4 ELEMENTS OF ALGEBRA. LCHAP, X. 

to wnich the ordinary process may be applied. 



4a2^2 _ 4^2^-2 



2ad — c2 



2ad-{-c^ 



+ 2ac'^d — Aa^c^ 



After having multiplied the dividend by 4a2, and performed 
th€ division, we obtain a remainder — 4a2c2 --(- c*, independent of 
the letter a : hence, the two polynomials, d? — c^ and 2ad — c^, 
are prime with respect to each other. Therefore, the greatest 
common divisor of the proposed polynomials is a — c. 

261. It sometimes happens that one of the polynomials cou 
tains a letter which is not contained in the other. 

In this case, it is evident that the greatest common divisor is 
independent of this letter. Hence, by arranging the polynomial 
which contains it, with reference to this letter, the required com- 
rdOH divisor will be the same as that which exists between the co- 
efficients of the different powers of the principal letter and iht 
second polynomial. 

By this method we are led, it is true, to determine the great 
est common divisor between three or more polynomials. But 
they will be more simple than the proposed polynomials. It 
often happens, that some of the co-efficients of the arranged 
^H>lynomial are monomials, or, that we can discover by simple 
inspection that they are prime with respect to each other ; and, 
\\, this case, we are certain that the proposed polynomials are 
pFime with respect to each other. 

Thus, in the example of the last article, after having suppressed 
fclie common factor a ~ c, which gives the results, 



dp- — c^ and 2ad — c^. 



we know immediately that these two polynomials are prime with 
respect to each other ; for, since the letter a is contained in the 
second and not in the first, it follows from what has just been said, 
that the common divisor must be contained in the co-efficients 2d 



CHAP. X.J GREATEST COMMON DIVISOR. 325 

and — c2 ; but these are prime with respect to each 3ther, and 
consequently, the expressions d"^ — c^ and 2ad — c^, are also prime 
with respect to each other. 

Let it be required to find the greatest common divisor of the 
two polynomials^ 

Zhcq + 30mjo + 186c + hmpq^ 
and, Aadq — 42^ + 24ac? — Ifgq. 

Now, the letter h is found in the first polynomial and not in 
the second. If then, we arrange the first with reference to A, 
we have, ^ 

(Zcq + 18c) 6 + ZOmp + ^mpq, 

and the required greatest common divisor will be the same as 
that which exists between the second polynomial and the two 
co-efiicients of J, which are, 

3cg 4- 18c and Z^mp -f 5mpq. 

Now, the first of these co-efiicients can be put under the form 
3c(<7 + 6), and the other becomes 6mp{q + 6) ; hence, q -\- 6 is 
a common factor of these co-efiicients. It will therefore be 
sufficient to ascertain whether g + 6 is a factor of the second 
polynomial. 

Arranging this polynomial with reference to q^ it becomes 
{4ad - lfg)q - 42fg -f 24ac^ ; 

and as the second part, 24c?(i — 42fg =zQ[4ad — *7fg), it follows 
that this polynomial is divisible by g' + 6, and gives the quotient 
4ac? — 7fg. Therefore, g + 6 is the greatest common divisor of 
the proposed polynomials. 

EXAMPLES. 

1. Pind the greatest common divisor of the two polynomial? 

6a;5 _ 4;c4 -Ux^- Sx^ - 3a? — 1, 

and 4a;* + 2x^ — IS.t^ + Sx — 5. 

Ans 2x^ — 4a;2 -^ x — I 



826 ELEMENTS OF ALGEBRA. [CHAP. X. 

2. Find the greatest common divisor of the polynomials 
20.2;6 - 12x^ + IQx^ _ 15^3 ^ X4a;2 - 15a; + 4, 
ftnd 15a:* - 9x^ -\- 47a;2 _ 21a; + 28. 

Ans, 5a;2 — Sx -\- 4, 
8. Find the greatest common divisor of the ♦two polynomials 
5a*^2 _j_ 2a3i3 _^ ^^2 _ 3^254 _|. ^^^^ 

«id ^ a^ + 5a=^£/ — a^'^ + Sa^^,^/. 

Transformation of E^iuations. 

262s The object of a transformation, is to change an equation 
from a given form to another, from which we can more readily 
determine the value of the unknown quantity. 

First, 

To chartge a given equation involving fmctional co-efficients to another 
of the same general form ^ hut having the co-efficients of all its terms entire 
If we have an equation of the form 

y 

and make x =z -r-\ 

k 

in which y is a new unknown quantity, and k entirely arbitrary; 
we shall have, after substituting this value for a:, and multiplying 
every teira by A:*", 

y^ 4- Pkij-^-^ + Qk'^y^-'^ + MPy""-^ + . . . + Tk'^-^y -f Uk'^ - 0, 

an equation in which the co-efficients of the different powers of 
y are equal to those of the same powers of x in the given equa- 
tit)n, multiplied respectively by k^, k^, k"^, k^, k^, &;c. 

It is now required to assign such a value to k as will make 
lije CO efficients of the different powers .of y entire. 

To illustrate, let us take, as a general example, the equation 

* X* f ^x^ -f 4-^' + 4^ + T- = ^» 

a J h 



CHAF X.] TRANSFOKMATION OF EQUATIONS. 327 

which becomes, after substituting ^ for x, and multiplying by k\ 

Now, there may be two cases — 

1st. Where the denominators i, c?, /, A, are prime with respect 
to each other. In this case, as k is altogether arbitrary, take 
h = hJfh^ the product of the denominators, the equation will then 
become, 

yA. 4_ adfh . y3 -}- ch'^dfVi^ . 1/ + ebH'^f-h^ . y + g¥d'^f^h^ =0, 

in which the co-efficients of y are entire, and that of the first 
term is 1. 

2d. When the denominators contain common factors, we shall 
evidently render the co-efficients entire, by making k equal to the 
least common multiple of all* the denominators. But we can 
simplify still more, by giving to k such a value that ^^, A-^, A:^, . . . 
shall contain the prime factors which compose 6, <i, /, A, raised 
to powers at least equal to those which are found in the de^ 
nominators. 

Thus, the equation 



becomes 



"^ ~ir^' ■*"12'' ~ 150 ''" 9000-"' 



'--6-^ +12"^ -150^-9000 = ^' 



y 

after making x = — , and reducing the terms, 

fC 

First, if we make k = 9000, which is a multiple of all the 
other denominators, it is clear that the co-efficients become entire 
numbers. 

But if we decom.pose 6, 12,^150, and 9000, into their prime 
factors, we find, 
6 = 2x3, 12 = 22x3, 150 = 2x3x52, 9000 = 2-^ K 32 x 5'>» ; 

and by making 

A = 2 X 3 X 5, 



828 ELEMENTS OF ALGEBRA. [CHAP. :X 

the product of the different prime factors, we obtain 

^■2 = 22 X 32 X 52, A;3 = 23 X 33 X 53, A;* = 2* X 34 X 5* ; 
>» hence we see that the values of k, k"^, k^, k'^, contain the 
prime factors of 2, 3, 5, raised to powers at least equal to 
those which enter into 6, 12, 150, and 9000. Hence, making 

^ = 2x3x5, 
is sufficient to make the denominators disappear. Substituting 
this value, the equation becomes 

^ 5.2.3.5 ^ 5.22.32.52 ^ 7.23.33.53 13.2^3^.5-^ _ 

^ 2.3 ^ "* 22.3 ^ ■ 2.3.52 y . 23.32.53 ~ ' 

which reduces to 

y4 _ 5 5^3 ^ 5.3.52^2 _ 7.22. 32. 5y - 13.2.32.5 = ; 
or, y* — 25?/3 -I- 375y2 _ UQOij - 1 170 ^ 0. 

Hence, we perceive the necessity of taking k as small a 
number as possible : otherwise, we should obtain a transformed 
equation, havii^ its co-efficients very great, as may be seen by 
reducing the transformed equation resulting from the supposi- 
tion k = 9000. 

Having solved the transformed equation, and found the values 

of y, the corresponding values of x may be found from the 

y 
equation, x =: — , 

by substituting for y and k their proper values. 

EXAMPLES. 

7 , 11 25 ^ 

1. ^'-T^' + 36-^-72 = ^ 

Making a; =— , and we have, 
o 

y3 _ Uy2 _|_ l]y _75 ^0. 
1^4_L?l3_^2 _!?_ L_n 

2. ^'-T2'^+40'' 225"^ 600 "" 800 ~ ^• 
Making X =: ^^ = ^, and we have, 

y5 „ 652/* + 1890y3 - 30720?/2 _ 928800y - 972000 = 0. 



CHAP. X.] 



TKAJSrSFORMATION OF EQUATIONS. 



329 



Second. 

To make the second or any other term disappear from an 
equation. 

26 3 • The difficulty of solving an equation generally diminishes 
with the number of terms involving the unknown quantity. 

Thus the equation 

x'^ = q, gives immediately, a; = ± Vg^ 
whi'je the complete equation 

x^ + 2px -^ q z=0, 
requires preparation before it can be solved. 

Now, any given equation can always be transformed into an 
incomplete equation, in which the second term shall be wanting. 

For, let there be the general equation, 

Suppose X ^^u -\- x'., 

u being a new unknown quantity, and x' entirely arbitrary. 

By substituting u -{- x^ for x, we obtain 
{u 4- xY +JP{^^ + ^)"^' + Q{u + x'Y-'^ . . . + T{u + x') + f7 = 0. 
' Developing by the binomial formula, and arranging with refer- 
ence to Uy we have 

m. — ■■ 1 

2 + . . . + a:""« 



v = o. 



ym _j_ ^lyf 


m — \ _ 
umr-i^rn\-——x''^ 


+ p 


+ (m- l)Px' 




+ ^ 



_j_ Px^mr-l 

+ Qx""^"^ 
4- . . . 

+ Tx' 

Since 7/ is entirely arbitrary, we may dispose of it in such 
way that we shall have 



m«^ -\- P = 0\ whence, xf 



P 



380 ELEMENTS OF ALGEBRA. [CHAP X 

Substituting this value of x^ in the last equation, we shall 
obtain an incomplete equation of the form, 

ym _|_ qu'^-^ _|- B/u"^--^ + . . . Tu+ U' = 0, 

in which the second term is wanting. 

If this equation were solved, we could obtain any value of 
X corresjDonding to that of u, from the equation 

F 

X =z u -\- x\ since x =■ u . 

m 

We have, then, in order to make the second term of du 
equation disappear, the following 

RULE. 

Substitute for the unknown quantity a new unknown quantity 
minus the co-efficient of the second term divided by the exioonent 
which exjjr esses the degree of the equation. 

Let us apply this rule to the equation, 

x"^ + 2/?.-? =z q. 
If we make x — u — p, 

we have {u — pY -\- 2p {u — p) = q', 

and by performing the indicated operations and transposing^ 
we find 

• 2/2 _ ^2 _|_ g^ 

263*. Instead of making the second term disappear, it may 
be required to fmd an equation which shall be deprived of its 
third, fourth, or any other term. This is done, by making the 
co-efficient of w, corresponding to that term, equal to 0. 

For example, to make the third term disappear, we mako, 
in the transformed equation, (Art. 2G3), 

from which we obtain two values for x\ which substituted in 
the transformed equation, reduce it to the form, 

ym ^ pV"-i -f R'u"^'^ . . . -\- Tu + W = 0. 



CHAP. X.] OF DERIVED POLYNOMIALS. 331 

Beyond the third term it will be . necessary to solve an 
equation of a degree superior to the second, to obtain the value 
of x' ', and to cause the last term to disappear, it will be neces- 
sary to solve the equation, 

x'^ + Px'^'-^ ...-{- Tx' + U=0, 
which is what the given equation becomes when x' is sub- 
stituted for X. 

It may happen that the value, 

m 
which majies the second term disappear, causes also the disap 
pearance of the third or some other term. For example, in 
order that the third term may disappear at the same time 
with the second, it is only necessary that the value of x^^ 
which results from the equation, 

,_ _P 
~~ m' 
shall also satisfy the equation, 

m — -— x'^ -\- {m-l)Fx'+ Qz= 0. 

F 

Now, if in this last equation, we replace x'' by — — , we have 

W _ 1 P2 P2 

^!!!__il__(m-l)— -f ^ = 0, or (m-l)F^-2mQ = 0' 
and, consequently, if 

~m-V 
the disappearance of the second term will also involve that of 
the third. 

Formation of Derived Polynomials. 

264i That transformation of an equation which consists in 
substituting u -\- x' for re, is of frequent use in the discussion 
of equations. In practice, there is a very simple method of 
obtaining the transformed equation which results from this sub 
«*-,itution. 



832 



ELEMENTS OF ALGEBRA. 



LCHAP. X 



To show this, let us substitute for x, u -\- x' in the ec^uation 

then, by developing, and arranging the terms according to th« 
ascending powers of ■w, we have 



+ . . . 
+ Tx' 



-\-mx' 



-i-{m-l)Fx'^ 

+ {771-2) Qx^^' 
+ . . . 



u-[-m 



+(' 



m 



1.2 



^m-2 






+(^-^)-r7r^^ 



1.2 



+ 



^2 + 



^=.0. 



By examining and comparing the co-efficients of the ditierent 
powers of w, we see that the co-efficient of w^, is what the first 
member of the given equation becomes when a/ is substituted 
in place of x ; we shall denote tliis expression by X\ 

The co-efficient of v? is formed from the preceding term X\ 
by multiplying each term of X^ by the exponent of a/ in that 
term, and then diminishing this exponent by 1 ; we shall denote 
this co-efficient by Y\ 

The co-efficient of u^ is formed from Y^^ by multiplying each 
term- of Y^ by the exponent of a/ in that term, dividing the 
product by 2, and then diminishing each exponent by 1. Repre- 

senting this co-efficient by — , we see that Z' is formed from Y^ 

in the smve manner that Y^ is formed from X\ 

In general, the co-efficient t)f any power of w, in the above 
transformed equation, may be found from the preceding co-efficient 
in the following manner, \dz. : — 

Multiply each term of the preceding co-efficient hy the expojieni 
of x' in that term, and dirni?iish the exponent of xf hy 1 ; then 
divide the algebraic sum of these expressions hy the number of 'pre- 
ceding co-efficients. 



CHAP. X.J OF DERIVED POLYNOMIALS. 333 

The law by which the co-eflFicient§, 



^ 1 1 o' 



Z' F 



1.2' 1.2.3' 

are derived from each other, is evidently the same as that 
which governs the formation of the numerical co-efficients of 
the terms in the binomial formula. 

The expressions, Y\ Z\ P, W\ &c., are called successive de- 
rived polynomials of X\ because each is derived from the pre- 
ceding one by the same law that Y^ is derived from X\ 

Generally, any polynomial which is derived from another by 
the law jiist explained, is called a derived 'polynomial. 

Recollect that X' is what the given polynomial becomes when 
vf is substituted for x, 

T' is called the first-derived polynomial ; 
Z' is called the second -derived polynomial ; 
V is called the third-derived polynomial ; 
&c., &;c. ' 

We should also remember that, if we make w = 0, we shall 
have a/ = ar, whence X' will become the given polynomial, from 
which the derived polynomials will then be obtained. 

• 265« Let us now apply the above principles in the following 

EXAMPLES. 

1. Let it be required to find the derived polynomials of the 
first member of the equation 

3a;* + 6a;3 — 8a;2 -f 2a; + 1 = 0. 

Now, u being zero, and xf = ar, we have from the law of 
fbrmli'g ^he derived polynomials, 

X = 3a;* -h 6a;3 - 3a;2 -|- 2a; + 1 ; 
Y' = 12a;3 -f 18a;2 _ 6a; 4- 2 ; 
Z' =3ea-2-f 86.r -6; 
r' = 72a; +36; 
W' = 72. 



334 ELEMENTS OF ALGEBRA. [CHAP. X 

It should be remarked that the exponent of ar, in the terms 1, 2, 
~ 6, 36, and 72. is equal to ; hence, each of those terms 
disappears in the following derived polynomial. 

2. Let it be required to cause the second term to disappear 
in the equation 

x^ - 122;3 4- 17:c2 _ 9^ ^ 7 ^ 0. 

12 
Make (Art. 263), x ^u + — = u-hS; 

whence, a/ — 3. 

The transformed equation will be of the form 

and the operation is reduced to finding the values of the co- 
efficients 

Y' V^ — 

^' ' 2' 2.3* 

Now, it follows from the preceding law, for derived poly- 
nomials, that 
X' = (3)4-12.(3)3+17.(3)2-9.(3)^ + 7, or X^ =-110; 
Y' =4.(3)3-36.(3)2+34.(3)1-9, or - - F' =-123; 

^ =6.(3)2-36.(3)1 + 17, or ^ = ._ 37; 

o=^-(^)'-i^ 2ra = <'- 

Therefore, the transformed equation becomes , 

u'^ _ ziii^ - 123?^ - 110 = 0. 
8. Transform the equation 

4^3 _ 5^2 _|. 7-c _ 9 _ 

into another equation, the roots of which shall exceed those of 
the given equation by 1. 

Make, x z=.u — \', whence x' z=z — 1 : 

and the tiansformed equation will be of the form 

X^ + Y^u + ^r/2 + __1_,,3 ^0 ♦ 



CHAP. X.J DERIVED POLYNOMIALS. 335 

We have, from the principles established, 
X' = 4.(-l)^- 5.(- 1)2 + 7. (-1)1-9, or X" = -25; 
F^ = 12.(-l)2-10.(-l)i-l-7 - - - . r'=+29; 

|-=12. (-1)1-5 y =~1^> 

— - =4 - - -— = 4-4. 

2.3 2.3 ^ 

Therefore, the transformed equation is, 

^v? — Yiv?- + 29zf - 25 = 0. 

4. What is the transformed equation, if the second term be 
made to disappear from the equation 

x^ - IO2;* + 7ri;3 + 4ar- 9 = 0? 

Ans. u^ — 33^3 - 118w2 -152w - 73 = 0. 

5. What is the transformed equation, if the second term be 
made to disappear from the equation 

3a:M- 15:^2 + 25^- 3 = 0? 

Am. e,3 _ I^ ^ 0. 

27 

6. Transform the equation 

3a;* — 13a;3 + 7.^2 _ 8a^ — 9 = 
into another, the roots of which shall be less than the roots of 
the given equation by — . 

o 

Ans. 3w* — 9^3 __ 4^2 1. ^ _ ^ 

9 3 

Properties of Derived Polynomials. 

266c We will now develop some of the properties of derived 
polynomials. 

Let x"^ + Pa;"-i + Qx"^"^ . . . Tx -{- U = 

be a given equation, and a, b, c, d, &c., its m roots. We shall 
Uien^have (Art. 248), 

jpm -j_ Pa;m~l _|_ Q3fn-2 ^ ^ = (^x — o) {x — b) (x -^ c) . . . {x — I), 



836 ELEMENTS OF ALGEBRA. [CHAP. X. 

Making - x =z a/ -\- u, 

or omitting the accents, and substituting x -{- u for x, and we have 
{x + w)'" + F{x + w)'^-i + . . . =z {x + u — a) (x + u — b) . . ,; 
or, changing the order- of x and u, in the second member, and 
regarding x — a, x — 6, . . . each as a single quantity, 



{x -f- uY +^(^ 4- uY~^ ... ={u-\- x —a) {u-{-x—b) . . . {u+x—l). 
Now, by performing the operations indicated in the two 
members, we shall, by the preceding article, obtain for the first 
member, 

X+ Yu^—u'-^ . . . u^', 

X beifig the first member of the proposed equation, and P", Z, &c., 

the derived polynomials of this member. 

With respect to the second member, it follows from Art. 251: 
1st. That the term involving w*^, or the last term, is equal to 

the product {x — a) {x — h) . . . {x — I) of the factors of the 

proposed equation. 

2d. The co- efficient of u is equal to the sum of the products 
of these m factors, taken m — 1 and m — 1. * 

3d. The co-efficient of m^ is equal to the sum of the product? 
of these m factors, taken m — 2 and m — 2 ; and so on. 

Moreover, since the two members of the last equation are 
identical, the co-eflicients of the same powers of u in the two 
members are equal. Hence, 

X = {x — a) [x — b) {x — c) . . . {x — /), 
wliich was already shown. 

Hence, also, Y, or the first derived polynomial, is equal to the 
sum of the products of the m factors of the first degree in the pro- 
'posed equation^ taken m — 1 and m — 1 / or equal to the algebraic 
sum of all the quotients that can be obtained by dividing X bij 
each of the m factors of the first degree in the proposed equation ^ 
Uiai is, 

^=— + — .+ — + ---^- ' 

X ~ a X — X — c X — I 



i 



CHAP X.] EQUAL ROOTS. 387 

Also, — , that is, the second derived polynomial, divided by 2, 

vs equal to the sum of the products of the m factors of the fir»t 
member of the proposed equation^ taken m — 2 and m — 2 ; or 
equal to the surn of the quotients obtained by dividing X by each 
of tht different factors of the second degree ; that is, 

Z X X X 



2 {x — a){x—b) (x — a) [x — c) ' ' {x — k)[x — I) 
and so on. 

' Of Equal Boots. 

267. All equation is said to contain equal roots, when its first 
member contains equal factors of the first degree with respect to 
the unknown quantity. When this is the case, the derived poly- 
nomial, which is the sum of the products of the m factors takeu 
m — 1 and m — 1, contains a factor in its different parts, which 
is two or more times a factor of the first member of the pro- 
posed equation (Art. ,266) : hence, 

There must be a common divisor between the first member of the 
proposed equation, and its first derived polynomial. 

It remains to ascertain the relation between this common divi- 
sor and the equal factors. 

268. Having given an equation, it is required to discover whether 
it has equal roots, and to determine these roots if possible. 

Let us make 

X= x"" -{- Fx"^- -f Qx"^-^ -i- . . . + Tx-\- U=0, 

and suppose that the second member contains n factors equal to 
X — a, n^ factors equal to x — b, n'' factors equal to x — c . . ,^ 
and also, the simple factors x — p, x — q, .r — r . . . ; we sliall 
then have, 
X .-{X — ay {x — bY (« — cY' ... (a: — p) {x — q) {x — r) (1). 

We have seen that V, or the derived polynomial of X, is 
the sum of the quotients obtained by dividing X by each of the m 
factors of the first degree in the proposed equation (Art. 266). 

22 



388 ELEMENTS OF ALGEBRA. [CHAP. X. 

Now, since X contains n fixctors eqaal to x — «, we shall 

V 

have n partial quotients equal to ; and the sanle reason 

ing applies to each of the repeated factors, x — 5. x -- c 

Moreover, we can form but one quotient for each simple factor, 
which is of the form, 

XXX 



I 



X— px— qx— r 
therefore, the first derived polynomial is of the form, 

x — a X — x — c X — p X — q x — r 

By examining the form of the value of X in equation (1), 
it is plain that 

{x - a)«-i, {x - by'-\ {x - c)»"-i . . . 
are factors common to all the terms of the polynomial F"; 
hence the product, 

{x - ay-^ X [x- 5)»^-i X (x — c)«"-i . . . 
is a divisor of Y. Moreover, it is evident that it will alsc 
divide X: it is therefore a common divisor of X and Y; and 
it is their greatest common divisor. 

For, the prime factors of X, are .x —a, x — b, x —c . . ., and 

X —p, ^ — q, X — r . . . ; now, x — p, x — q, x — r, cannot 

divide Y, since some one of them will be wanting in some of 

the parts of Y, while it vdW be a factor of all the other parts. 

Hence, the greatest common divisor of .X and Y, is 

D — (x — a)"-i {x — by-^ {x — c)«''-i . . . ; that is, 

The greatest common divisor is composed of the product of those 
factors which enter two or more times in the given equation, each 
raised to a power less by 1 than in the primitive equation. 

269. From the above, we deduce the following method foi 
finding .the equal roots. 

To discover whether an equation, 

x=€r, 

iOontains any equal roots: 



OHAP. IX.J EQUAL ROOTS. 389 

Is,!. Form Y^ or the derived 'polynomial of X ; then seelc fot 
the greatest common divisor between X and Y. 

2d. If one cannot he obtained^ the equation has no equal roots^ 
or equal factors. 

If we find a common divisor i), and it is of the first degree, 
or of the form a" — h, make x — h = 0, whence x — h. 

We then tonclude, that the equation has two roots equal to h, 
and has but one species of equal roots, from which it may he 
freed by dividing X by (x — h)^. 

If D is of the second degree with reference to x^ solve tki 
equation D =z 0. There may be two cases ; the two roots wilj 
be equal, or they will be unequal. 

1st. When we find JD := {x — h)^, the equation has three roots 
equal to h, and has but one species of equal roots, from which 
it can be freed by dividing X by (x — h)^. 

2d. When D is of the form (x — h) [x — A'), the proposed 
equation has two roots equal to h, and two equal to h', from 
which it may be freed by dividing X by {x — hy (x — h')\ 
or by D"^. 

Suppose now that D is of any degree whatever ; it is necessary^ 
in oi'der to know the species of equal roots, and the number 
of roots of each species, to solve completely the equation^ 

D = 0. 

Then, every simple root of the equation I) =.0 will be twice a 
root of the given equation; every double root of the equation D =z 
will be three times a root of the given equation ; and so on. 

As to the simple roots of 

X=0, 

we begin by freeing this equation of the equal factors contained 
in it, and the resulting equation, X' = 0, will make known the 
simple roots. 



MO ELEMENTS OF ALGEBRA. [CHA.P. X. 

EXAMPLES. 

1. Determine whether the equation, 

2x^ — I2x^ + 19a;2 __ 6a; + 9 = 0, 

contains equal roots. 

We have for the first derived polynomial, 

8a;3 — 36.i;2 + 38a; — 6. 

Now, seeking for the greatest common divisor of these poly 
nomials, we find 

i> = a; — 3 = 0, whence x = Z: 
bence, the given equation has two roots equal to 3. 
Dividing its first member by {x — 3) 2, we obtain 

2x^-\-l =0] whence, x z= zt — V— 2. 

The equation, therefore, is completely solved, and its roots are 

3, 3, -fl/ZT^and -1^32. 

2. For a second example, take 

x^ — 2x^ + Sx^ — 7a;2 4- 8a; — 3 = 0. 
The first derived polynomial is 

5a;* — 8a*3 + 9a;2 — 14a; + 8 ; 
and the common divisor, 

a;2 - 2a; + 1 = (a; - 1)2 : 

hence, the proposed equation has three roots equal to 1. 
Dividing its first member by 

{a;-])3 = a;3-3a;2 + 3a;-l, 
^e quotient is 

a;2 + a; + 3 ~ ; whence, x = ^ — — ; 

thus, the equation is completely solved. 



CHAP. X.J EQUAL ROOTS. 841 

3. For a third example, tak<> the equation 

The first derived polynomial is 

Ix^ + ZQx^ + 30a;* — 24a;3 — ASix^ ~ 6a; -f 8 ; 
and the common divisor is 

x^ + 3a;3 + a;2 — 3a; — 2. 
The equation, 

a;* + 3a;3 -f a;2 — 3a; - 2 = 0, 
cannot be solved directly, but by applying the method of equal 
roots to |t, that is, by seeking for a common divisor between 
its first member and its derived polynomial, 

4a;3 4- Qx^ + 2a^ — 3 : 
we find a common divisor, x -{• \ \ which proves that the square 
of a; + 1 is a factor of 

a;* + 3a;3 + a;2 — 3a; — 2, 
and the cube of a; + Ij a factor of the first member of the 
given equation. 
Dividing 

x^ + 3a:3 -f a;2 — 3a; — 2 ^j (3. + 1)2 _ 3^2 _^ 2a; + 1, 
WQ have a;^ -f a; — 2, which being placed equal to zero, gives 
the two roots a; = 1, a; = — 2, or the two factors, a; — 1 and 
« + 2. Hence, we have 

a;* + 3a;3 -j- a;^ _ 3a; - 2 = (a; + 1)^ {x - 1) (a; + 2). 
Therefore, the first member of the proposed equation is equal to 
(a; +1)3 (a- -1)2 (a; + 2)2; 
that is, the proposed equation has three roots equal to — 1, two 
equal to +1, and two equal to — 2. 

4. What is the product of the equal factors of the equatioii 
£' — 7a;6 + lOa;^ + 22a;4 — 43a;3 — 35a;2 + 48.1; + 36 = ? 

Ans. (a; - 2)2 (;r- 3)2 (a; +1)3. 

5. What is the product of the equal factors in the equation, 

x^ - 3a;6 + 9a;5 - 19a;* + 27a;3 — 33a;2 + 27a; - 9 == ? 

Ans. (a;-l)3(a;2 + 3)^ 



842 ELEMENTS OF ALGEBRA. [CHAP. 

Elimination. 

870i We have already explained the methods of eliminating 
OLS unknown quantity from two equations, when these equations 
are of the first degree with respect to the unknown quantities. 

When the equations are of a higher degree than the first, 
die methods explained are liot in general applicable. In this 
case, the method of the greatest common divisor is considered the 
best, and it is this method that we now propose to mvestigate. 

One quantity is said to be a function of another when it de- 
pends upon that other for its value; that is, when the quan- 
tfities are so connected, that the value of the latter cannot be 
changed ^vithout producing a corresponding change in the former. 

27 1 • If two equations, containing two unkno\\Ti quantities, be 
combined, so as to produce a single equation containing but one 
imknown quantity, the resulting equation is called a final equa- 
tion ; and the roots of this equation are called compatible 
values of the unknown quantity which enters it. 

Let us assume the equations, 

P = and ^ r= 0, 

in which P and Q are functions of x and y of any degree 
whatever ; it is required to ccjmbine these equations in such a 
manner as to eliminate one of the unkno^\Ti quantities. 

If we suppose the final equation involving y to be found, and 
that y = a is a root of this equation, it is plain that this value 
of y,. in comiection with some value of a;, will satisfy both 
equations. 

If then, we substitute this value of y in both equations, there 
will result two equations contaming only rr, and these equations 
will have at least one root in common, and consequently, their 
first members will have a common divisor involving x (Art. 246). 

This common divisor will be of the first, or of a higher degree 
with respect to ar, according as the particular value of y = a cor 
responds to one or more values of x. 




CHAP. XI.j ELIMINATION. 

Conversely, every value of y ivhich, being substituted in the 
two equations^ gives a common divisor involving x, is necessarily 
a compatible value, for it then satisfies the two equations at the 
same time with the value or values of x found from this common 
divisor w^hen put equal to 0. 

272. We will remark, that, before the substitution, the first 
members of the equations cannot, in general, have a comvion divi- 
sor which is a function of one or both of the unknown quantities. 

For, let us suppose, for a moment, that the equations 

P = and ^ = 0, 

are of the form 

P^ X i2 = and ^^ X i2 = 0, 
R being a fujjction of both x and y. 

Placing P = 0, we obtain a single equation involving two 
unknown quantities, which can be satisfied with an infinite number 
of systems of values. Moreover, every system which renders R 
equal to 0, would at the same time cause P' . R and Q' . R to 
become 0, and consequently, would satisfy the equations 
P =: and ^ = 0. 

Thus, the hypothesis of a common divisor of the two poly- 
nomials P and Q, containing x and y, brings with it, as a con- 
sequence, that the proposed equations are indeterminate. There- 
fore, if there exists a common divisor, involving x and y, of the 
two polynomials P and Q, the proposed equations will be inde- 
terminate, that is, they may be satisfied by an infinite number 
of systems of values of x and y. Then there is no data to 
determine a final equation in y, since the number of values of y 
is infinite. 

Again, let us suppose that P is a function of x only. 

Placing R = 0, v/e shall, if the equation be solved with 
reference to x, obtain one or more values for this unknown 
quantity. 

Each of these values, substituted in the equations 
P' . i2 = and ^^ P = 0, 



344 ELEMENTS OF ALGEBRA. [CHAP. X 

will satisfy them, whatever value we mav attribute to y, since 
these values of x would reduce R to 0, independently of y. 
Therefore, in this case, the proposed equations admit of a ^nite 
number of values for x^ but of an inlinite number of values for 
y and then, therefore, there cannot exist a final equation in y. 
Hence, when the equations 

are determinate, that is, when they admit only of a limited 
number of systems of values for x and y, their first members 
cannot have for a common divisor a function of these unknown 
quantities^ unless a particular substitution has been made for one 
of these quantities. 

273i From this it is easy to deduce a process for obtaining 
the final equation involving y. 

Since the characteristic property of every compatible value 
of y is, that being substituted in the first members of thQ two 
equations, it gives them a common divisor involving x, which 
tiiey had not before, it follows, that if to the two proposed 
polynomials, arranged with reference to x, we apply the process 
for finding the greatest common divisor, we shall generally not 
find one. But, by continuing the operation properly, we shall 
arrive at a remainder independent of x^ but which is a function 
of ?/, and which, placed equal to 0, will give the required final 
equation. 

For, every value of y found from this equation, reduces to 
zero the last remainder in the operation for finding the common 
divisor ; it is, then, such that being substituted in the preceding 
remainder, it will render this remainder a common divisor of the 
first members P and Q. Therefore, each of the roots of the 
equation thus formed, is a compatible value of y. 

274» Admitting that the fmal equation may be completely 
solved, which v.'ould give all the compatible values, it would 
afterward be necessary to obtain the corresponding values of x. 
Now, it is evident that it would be sufficient for this, to sub- 
stitute the different values of y in the remainder preceding the 



t 



CHAP. X.] ELIMINATION. 845 

last, put the polynomial involving x which results from it, equal 
to 0, and find from it the values of x\ for these polynomials 
are nothing more than the divisors involving a;, which become 
common to A and B. 

But as the final equation is generally of a degree superior to 
the second, we cannot here explain the methods of finding the 
values of y. Indeed, our design was principally to show that, 
two equations of any degree being given, we can, without supposing 
the resolution of any equation^ arrive at another equation^ contain- 
ing only one of the unknown quantities which enter into the pro* 
posed equations. 

EXAMPLES. 



1. Having given the equations 




x'^ + xy +y''- 1=0, 




:r3 -f y3 ^ 0, 




to find the final equation in y, . 




First Operation. 




x' + y^ 1 a;2 + a;y + 2/^ - 


-1 


x^ ^yx^ + {y'-^)x X _y 




-yar2 - (2/2 _l)rr 4-^3 




— yx"^ — y'^x — y3 -j_ y 





X -\- 2y^ — y = 1st remainder. 

Second Operaiicm. 

ic2 -f yx +2/2_i I I re + 2y^ — y 

x2 -f (2y3 _ y)^ \\ ^ _ ^2^3 _ 2^j 

- (2ij^ - 22j) X -\- y2_i 
~ (2^/3 -2y)x- 4y6 + Qy* - ^t/^ 
4/ _ 6y* + 3y2 _ 1. 

Hence, the final equation in y, is 

%« — 6?/4 + 3y2 _ 1 = 0. 



B4:Q ELEMENTS OF ALGEBRA. [CH^P. X. 

If it were required to find the final equation in u;, we observe 
that X and y enter into the primitive equations under the same 
forms ; hence, x may be changed into y and y into re, without 
destroying the equality of* the members. Therefore, 
4^6 _ 6^.4 + 3^.2 _ 1 — 

is the final equation in x. 

2. Find the final equation in y, from the equations 
x3 _ St/x^ 4- (3/ -1 y + 1) a; - y3 + 3^-2 _2y ^ Q, 
x^ — 2yx -\- y^ — y z= 0. 

First Operation. 
^ - Syx^ 4- (3y2 - y ^ l)x - y^ + y^ - 2y\ \x^ - 2xy + y^ -j ^ 
z^ — 2yx'^ -{- [y^ — y)x x — ;y 

- y^2_|.p^2_^l)^_^3^_ y2 ^ 2y 

— .ya;2 -|- 2y'^x — y^ + y"^ 



^ 



x-2y 


Second, Operation. 


x^ — 2xy-\-y'^ — y 


x-2: 


x^ — 2xy 


X 


y"- -y 




2/' - 2/ - 0, 





Hence, 

is the final equation in y. This equation gives 
y ^=.1 and y = 0. 
Placmg the preceding remainder equal to zero, and substi- 
tuting therein Xho, valaes of y, 

y = 1 and y = 0, 
we find for the corresponding values of a:, 
X :=^2 and re r= ; 
from which *^' ^iven equations may be entirely solved. 



CHAPTEK XI. 

SOLUTION OF NUMER OAL EQUATIONS CONTAINING BUT ONE UNKNOWS 

QUANTITY. — Sturm's theorem. — cardan's rule. — horner's method. 

275* The principles established in the preceding chapter, are 
applicable to all equations, whether the co-efficients are numerical 
or algebraic. These principles are the elements which are em- 
ployed in the solution of all equations of higher degrees. 

Algebraists have hitherto been unable to solve equations of a 
higher degree than the fourth. The formulas which have been 
deduced for the solution of algebraic equations of the higher 
degrees, are so complicated and inconvenient, even when they 
can be applied, that we may regard the general solution of an 
algebraic equation, of any degree whatever, as a probleija more 
curious than useful. 

Methods have, however, been found for determining, to any 
degree .of exactness, the values^ of the roots of all numerical 
equations ; that is, of those equations which, besides the unknown 
quantity, involve only numbers. 

It is proposed to develop these methods in this chapter. 

276. To render the reasoning general, we will take the 
equation, 

X =: a;"* -f Po;"^! + Qx-^-"^ + . . . U — 0. 
in which P, ^ . . . denote particular numbers which are real, 
and either positive or negative. 

If we substitute for x a number a, and denote by A what 
X becomes under this suppositi on ; and again substitute a -{- u 
for ar, and denote the new polynomial by A^ : then, u may he 
taken so small, that the difference between A' and A shall be 
less than any assignable quantity. 



348 ELEMENTS OF A.LGEBIIA. [CHAP. XI, 

If. no^, vre denote by ^, C, D, . . . . wliat the co-efficients 

F, — , - — - (Art. 264), become, vrh^n we make x = c, we 

shall have, 

A' = A ^ Bu-\- Cu^ + 2)i^3 -f . . . -t- '^^ - - . (1) ; 
whence, 

A' — A = Bu-\- Cu^ 4- i)t*3 +...+«*"• . - - (2). 

It is now required to show that this difference may be ren- 
dered less than any assignable quantity, by attributing a value 
sufficiently small to u. 

If it be required to make the difference between A'' and A 
less than the number N. we must assign a value to u which 
will satisfy the inequality 

Bu -f- Cu^ + Bu^ + u"'<:]sr . . . (3). 

Let us take the most unfavorable case that can occur, viz., 
let us suppose that every co-efficient is positive, and that each 
is equal to,- the largest, which we will designate by £'. Then 
any value of ic which will satisfy the inequality 

K{u -j-u^-\-u^+ 'w'")< iV^ - - - (4), 

will evidently satisfy inequality (3). 

Now, the expression within the parenthesis is a geometrical 
progression, whose first t^rm is u, whose last term i| w*", and 
whose ratio is u ; hence (Art. 188), 

o.Q, u — u u — u u _ . 

U -\- U^ -\- U^ -{-... lO^ = = r =- X (1 — W«). 

u — \ 1 — u 1 — u ^ ' 

Substituting this value in inequality (4), we have, 

^-^(l-n-'Xi^ .... (5). 

N 
\S now we make u = -— , the first factor of the first mem 
J.y "p xt 

— -J is less 

than 1. the second factor is less than 1 ; hence, the fiist mem 
ber is less than AT. 



CHAP. JCI.J NUMERICAL EQUATIONS. 349 

JSf 
We conclude, therefore, that u = -— --, and every smaller 

value of w, will satisfy the inequalkies (3) and (4), and conse- 
quently, make the difference between A' and A less than any- 
assignable number JV. 

If in the value of A\ equation (1), we make w= — , it 

Is plain that the sum of the terms 

Bu + Cv? -f Dv? + . . . w*" 
will be less than A^ from what has just been proved ; whence 
we conclude that 

In a series of terms arranged according to the ascending powers 
of an arbitrary quantity, a value may be assigned to that </ 
so small, as to make the first term numerically greater than the 
sum of all the other terms. 

m 

First Principle, 

277t If tf^o numbers p and q, substituted in succession in the 
place of X in the first member of a numerical equation, give results 
'affected with contrary signs, the proposed equation has a real root, 
comprehended between these two numbers. 

Let us suppose that p, when substituted for x in the first 
member of the equation 

X=zO, gives -\- B, 
and that q, substituted \n. the first member of the equation 
X = 0, gives —E\ 

Let us now suppose x to vary between the values of p and q 
by so small a quantity, that the difference between any two 
corresponding consecutive values of X shall be less than any- 
assignable quantity (Art. 276), in which case, we say that X is 
subject to the law of continuity, or that it passes through all 
the intermediate values between M and — B'. 

Now, a quantity which is constantly finite, and subject to the 
*aw of continuity, cannot change its sign frcm positive to nega 



850 ELEMENTS OF ALGEBRA. [CHAP. XI. 

tive, or from negative to positive, without passing through zero : 
hence, there is at least one number between p and q which will 
satisfv the equation 

and consequently, one root of the equation lies between these 
numbers. 

278 • We have shown in the last article, that if two numbers 
be substituted, in succession, for the unknown quantity in any 
equation, and give results affected with contrary signs, that there 
will be at least one real root comprehended between them. We 
are not, however, to conclude that there may not be more than 
one ; nor are we to infer the converse of the proposition, viz., 
that the substitution, in succession, of two numbers which include 
roots of the equation, will necessarily give results affected with 
contrary signs. 

Second Principle. 

279. When an uneven number of the real roots of an equation 
is comprehended between two numbers, the results obtained by sub- 
stituting these numbers in succession for x in the first member, will 
have contrary signs ; but if they comprehend an even number of 
roots, the results obtained by their substituiion will have the same sign. 
To make this proposition as clear as possible, denote by 
a, b, c, . . . those roots of the proposed equation, 

X=0, 
which are supposed to be comprehended between p and q, and 
by Y, the product of the factors of the first degree, with refer- 
ence to X, corresponding to the remaining roots of the given 
equation. 

The- first member, X, can then be put under the form, 

{x — a){x — b)(x~c)... X F=0. 
Now, substituting p and q in place of x, in the first mem. 
Der, ve shall obtain the two results, 

(p-a){p-b){p-c) ... XT', 
(g-a)(q-b)(q-c) . . kT". 



CHAP. XI.] NUMERICAL EQUATIONS. 35l 

Y^ and !P^ representing what Y becomes, when we replace in 
succession, a; by ^ and q. These two quantities Y' and Y^'^ are 
affected with the same sign; for, if they were not, by the first 
principle there would be at least one other real root com- 
prised between 'p and g, which is contrary to the hypothesis. 
To determine the signs of the above results more easily, 
divide the first by the second, and we obtain 

(jp — g) (jt? — 6) (jp — c) . . . X Y' 

which can be written thus, 

p — a p — b p — c Y' 

X 7 X X . • • -rr//' 

q — a q — q — c V 

Now, since the root a is comprised between p and q^ that 
is, is greater than one and less than the other, p — a and 
f — a must have contrary signs ; also, p — h and q — 5 must 
have contrary signs, and so on. 

Hence, the quotients 

p — a p — h p — c 
q — a q — o q — c 

are all negative. 

Y' 
Moreover, -— - is essentially positive, since Y' and Y" are 

affected with the same sign ; therefore, the product 
p — a p — h p — c Y' 

X 7 X X ... -xriti 

q — a q — b q — c x 

will be negative^ when the number of roots, a, &, c . . ., com 
prehended between p and q^ is uneven, and positive when the 
number is even. 

Consequently, the two results, 

{p-a){p-h)(p ^c) . . . X Y', 
and {^q — a) {q — b) (q — c) . . . X F", 

will have contrary signs when the number of roots comprised 
between p and q is uneven, and the same sign when the num- 
ber is even 



S52 ELEMENTS OF ALGEBKA. [CHAP. XI. 

Third Principle. 

280» If the signs of the alternate terms of an equation, he 
changed^ the signs of the roots will be changed. 

Take the equation, 

^m _|. p^m-\ _|_ Q^r^2 . . . -f f^ = Q - - (1) ; 

and by changing the signs of the alternate terms, we have 
^m _p^^i _|_ Q^m-2 . . . ± cr_ - - (2), 

or, — a;"* + Pa;'"-i — Qx'^'^ . . . ^ U = • - (3). 

But equations (2) and (3) are the same, since the sum of th« 
positive terms of the one is equal to the sum of the negative 
terms of the other, whatever be the value of x. 

Suppose a to be a root of equation (1); then, the substitution 
of a for X will verify that equation. But the substitution of 
— a for X, in either equations (2) or (3), will give the same 
result as the substitution of + a, in equation (1) : hence — a, 
is a root of equation (2), or of equation (3). 

We may also conclude, that if the signs of all the terms 
be changed, the signs of the roots will not be altered. 

Limits of Real Roots. 

281. The different methods for resolving numerical equations, 
consist, generally, in substituting particular numbers in the pro- 
posed equation, in order to discover if these numbers verify it, 
or vfhether there are roots coniprised between them. But by 
reflecting a little on the composition of the first member ot 
the general equation, 

a;m _|. p^m-\ J^ Qyfi-2 _ ^ ^ ^Ij ^ f^ — 0, 

we become sensible, that there are certain numbers, above which 
it would be useless to substitute, because all numbers above a 
certain limit would give positive results. 



CHAP. XI. i LIMITS OF KEAL ROOTS. 353 

282. It is now required to determine a number, which being 
substituted for x in the general equation, will render the first term 
x°» greater than the arithmetical sum of all the other terms ; 
tliat is, it ife required to find a number for x which will render 

Let k denote the greatest numerical co-efficient, and substitute 
il in place of each of the co-efficients; the inequality will then 
become 

Xm y ].^mr-\ j^ ]^^m-1 _^ _ ^ _j_ ^^ _|_ ^, 

\ 

It is evident that every number substituted for x which will 
satisfy thik condition, will satisfy the preceding one. Now, 
dividing both members of this inequality by ic"*, it becomes 

Making x — h, the second member reduces to 1 plus the 
sum of several fractions. The number h will not therefore 
satisfy the inequality; but if . we make xz=.h-\-\, we obtain 
for the second member the expression, 



This is a geometrical progression, the first term of which is 

k k 1 

^he last term, , and the ratio, - — -— ; hence, 



yt+ 1' ' {k^Xy ' k^V 

the expression reduces to 

k k 

(^ + 1)^+1 k ^\ 1__ 

^ + 1 

which is evidently less than 1. 

Now, any number > (-^ + 1), put in place of x, will render 

k k 

the sum of the fractions 1 -f . . . still less : therefore, 

x x^ 

The greatest co-efficient plus 1, or any greater number, being 
mdistituted for x, will render the first term x™ greater than tlu 
arithmetical sum of all the other terms. 

23 



854 ELEMENT* OF ALaEBRA. fCHAP. XI. 

283. Every number which exceeds the greatest of the positive 
roots of an equation, is called a superior limit of the positive roots. 

From this definition, it follows, that this limit is susceptible 
of an infinite number of values. For, when a number is found 
to exceed the greatest positive root, every number greater than 
this, is also a superior limit. The term, however, is generally 
applied to that value nearest the value of the root. 

Since the greatest of the positive roots will, when substituted 
for X, merely reduce the first member to zero, it follows, that 
we shall be sure of obtaining a superior limit of the positive 
roots by finding a number, which substituted iti place of x, renders 
the first member positive, and which at the same time is such, thai 
every greater number will also give a positive result; hence. 

The greatest ,co-ejficient of x plus 1, is a superior limit of 
(he positive roots. 

Ordinary Limit of the Positive Roots. 

284. The limit of the positive roots obtained in the last article, 
is commonly much too great, because, in general, the equation 
contains several positive terms. We wdll, therefore, seek for a 
limit suitable to all equations. 

Let a;*""" denote that power of x that enters the first nega- 
tive term which follows x'^, and let us consider the most unfavor- 
able case, viz., that in which all the succeeding terms are negative, 
and the co-efficient of each is equal to the greatest of the nega- 
tive co-efficients in the equation. 

Let aS^ denote this co-efficient. What conditions will render 

Dividing both members of this inequality by x'^, we hav(« 
. S . S ' S S S 

Now, by supposing 

X = \/ S -{- 1, or for simplicity, making "/o=r S \ 
<which gives, S = 6^'", and x = S' -^ 1, 



CHAP. XI.] LIMITS OF POSITIVE ROOTS. 855 

the second member of the inequality will become, 



wliich is a geometrical progression, of which is thfl 

first term, and the ratio. Hence, the expression for the 

/b + 1 

sum of all the terms is (Art. 188), 

(S'-h 1)^+1 ^^ ^, .^ _ ^ ^ 



^^+1 



(S'+l)'' S'^-^ 


^/«-l 


-(^'+1)"-^ 


{S' + 1)" 



Moreover, every number > aS^ + 1 or 11/ ;S^ + 1, will, when 
substituted for x^ render the sum of the fractions 

still smaller, since the numerators remain the same, while the 
denominators are increased. Hence, this sum will also be less. 

Hence, 'i/ aS + 1, and every greater number, being substituted 
for X, will render the first term x^ greater than the arithmetical 
sum of all the negative terms of the equation, and will conse 
quently give a positive result for the first member. Therefore, 

That root of the numerical value of the greatest negative co-effi- 
cient whose index is equal to the number of terms which precede 
the first negative term^ increased by 1, is a superior limit of the 
jiositive roots of the equation. If the co-efficient of a term is 0, 
the term must still be counted. 

Make n r= 1, in which case the first negative term is the 
second term of the equation ; the limit becomes 

that is, the greatest negative co-efficient plus 1. 

Let n =2; then, the limit is y^4- 1. When «» z= 3. the 
limit is 3/^+ 1. 



356 ELEMENTS OF ALGEBRA. ,'CHAP. XL 



EXAMPLES. 



1. What is the superior limit of the positive roots of rhe 
elation 

X* — 5a;3 -{- 87a;2 _ 3a; + 39 = 01 



Ans. y^ 4- 1 = 1/5 + 1 = 6. 



2. What is the superior limit of the positive roots of tlio 
equation 

x5 ^7x*— 12.^3 - ^9x^ + 522? - 13 = 01 



A71S. y^+ 1 =,y/^+ 1 = 8. 



3. What is the superior limit of the positive roots of the 
equation 

a;*-f lla;2-25a;- 67 = 0? 

In this example, we see that the second term is wanting, that 
is, its co-efficient is zero ; but the term must still be counted in 
fixing the value of n. We also see, that the largest negative 
co-efficient of x is found in the last term where the exponent of 
X is zero. Hence, 

"/^-h 1 = 3^67 + 1 ; 

and therefore, 6 is the least whole number that will certainly 
fulfil the conditions. 

Smallest Limit in Entire Numbers. 

285. In Art. 282, it was shown that the greatest co- efficient 
of X plus 1, is a superior limit of the positive roots. In the 
last article we found a limit still less ; and we now propose to 
find the smallest limit, in whole numbers. 

Let X = 



be the proposed equation. If in this equation we make 2; = r' -f-* 
xf being arbitrary, we shall obtain (Art. 264), 

X' + T'u + |-w2 -f . . . -f ?^- = (1). 



CHAP. XL] LIMITS UF POSITIVE ROOTS. 8df 

Let us suppose, that after successive trials we have determined 
a number for a/, which substituted in 

renders, at the same time, all these co-efficients positive, this nuim 

ber will in general be greater than the greatest positive root 

of the equation 

X = 0. 

For, if the co efficients of equation (1) are all positive, no 
positive value of u can satisfy it ; therefore, all the real values 
of u must\ be negative. But from the equation 

X =z x' + u^ we have u z= x —• x^] 
and in ordei that every value of u, corresponding to each of the 
values of x and a/, may be negative, it is necessary that the 
greatest positive value of x should be less than the value of x\ 
Hence, this value of x' is a superior limit of the positive 
roots. If we now substitute in succession for x m X the values 
x' — 1, x' — 2, x' — 3^ &;c.. Until a value is found which will 
make X negative, then the last number which rendered it posi- 
tive will be the least superior limit of the positive roots in 
whole numbers. 

EXAMPLE. 

Let x^ — 5a;3 - Gx^ — 19.r + 7 = 0. 

As x^ is indeterminate, we may, to avoid the incon'venience 
of writing the primes, retain the letter x in the formation of 
the deri ved polynomials , and we have, 

X = x^- 5a-3 - 6:i-2 - 19a; + 7, 
F = 4a;3-15ar2-12a: -19, 

-- = 6a;2_15a; -6, 

y . . 

The question is now reduced to finding the smallest entire 
number which, substituted in place of a?, will render all of 
these polynomials positive. 



S68 ELEMENTS OF ALGEBRA. LCHAP. XI. 

It is lAvJa that 2 and every number > 2, will render the 
polynomial of the first degree positive. 

B.it 2, substituted in the polynomial of the second degree, 
gives a negative result ; and 3, or any number > 3, gives a 
positive result. 

Now, 3 and 4, substituted in succession in the polynomial 
of the third degree, give negative results ; but 5, and any 
greater number, gives a positive result. 

Lastly, 5 substituted in X, gives a negative result; and so 
does 6 ; for the first three terms, x^ — 5x^ — Qx"^, are equiva- 
lent to the expression x^ [x — 5) — 6x^, which reduces to when 
X = 6; but X z^7 evidently gives a positive result. Hence 7, is 
the least limit in entire numbers. We see that 7 is a supe- 
rior limit, and that 6 is not ; hence, 7 is the least limit, as 
above shown. 

2. Applying this method to the equation, 

x^ — Sx^- Sx^ — 25a;2 4- 4a; - 39 = 0, 
the superior limit is found to be 6. 

8. We find 7 to be the superior limit of the positive roots 
of the equation, 

a;5 _ 5^4 _ 13^3 4. i7a,2 _ 59 _ 0. 

This method is seldom used, except in finding incommeu- 
surable roots. 

Supe^rior Limit of Negative Roots. — Inferior Limit of Posi 
tive and Negative Boots. 

286. Having found the superior limit of the positive roots, 
it remains to find the inferior limit, and the superior and in- 
ferior limits of the negative roots, numerically considered. 

jFirsf, If, in any equation, 

X = 0, we make x = — , 

y 

we shall have a new equation J^ = 0. 

Since we know, from the relation a; = — , that the greatest 



CHAP. XI. CONSEQUENCES OF PllINCIPLES. 359 

positive va.ue of y in the new equation corresponds to the leaat 
positive value of x in the 'given equation, it follows, that 

If we determine the superior limit of the positive roots of the 
equation Y = 0, its reciprocal will be the inferior limit of tha 
positive roots of the given equation. 

Hence, if we designate the superior limit of the positive 
roots of the equation F=: by Z^, we shall have for the inr 

ferior limit of the positive roots of the given equation, -=^ 

Second^ If in the equation 

X = 0, we make x =: — ij, 
which gives the transformed equation ]P =^ 0, it is clear that 
the positive roots of this new equation, taken with the sign 
— , w^ill give the negative roots of the given equation; there- 
fore, determining by known methods, the superior limit of the 
positive roots of the new equation Y' = 0, and designating this 
limit by L^'', we shall have — L''^ for the superior limit, (nu- 
merically), of the negative roots of the given equation. 

Third, If in the equation 

X = 0, we make x = , 

we shall havei. the derived equation Y"^ = 0. The greatest posi- 
tive value of y in this equation will correspond to the least 
negative value (numerically) of x in the given equation. If^ 
then, we find the superior limit of the positive roots of the 
equation Y^^ = 0, and designate it by L^^'', we shall have the 

inferior limit of the negative roots (numerically) equal to — y^ 

Consequences deduced from the preceding Principles. 

First. 

• 287. Every equation in which there are no variations in the signSy 
that is, in ivhich all the terms are positive, must have all of its real 
roots negative; for, every positive number substituted for Xy will 
render the first member essentially positive. 



360 . ELEMENTS OF ALGEBRA. [CHAP. XL 

Second. 

288 » Every comjolete equation^ having its terms alternately posi 
live and negative, must have its real roots all positive ; for, every 
negative number substituted for x in the proposed equation, would 
render all the terms positive, if the equation be of an even de 
gree. and all of them negative, if it be of an odd degree. Hence, 
their sum could not be equal to zero in either case. 

This principle is also true for every incom'plete equation, in which 
there results, hy substituting — y for x, an equation having all its 
terms affected ivith the same sign. 

Third. 

289. Ever}^ equation of an odd degree, the co-efficients of which 
are real, has at least one real root affected with a sign contrary to 
that of its last term. 

For, let 

x^ + Px-^-^ -\- . . . Tx± U=0, 

be the proposed equation ; and first consider the case in which 
the last term is negative. 

By making x = 0, the first member becomes — JJ. But by 
giving a value to x equal to the greatest co-effi(^ent plus 1, or 
{K -{- 1), the first term x^ will become greater than the arith- 
metical sum of all the others (Art. 282), the result of this sub- 
stitution will therefore be positive; hence, there is at least one 
real root comprehended between and ^+1, which root is posi- 
tive, and consecjuently afiected with a sign contrary to that of the 
last term (277). 

Suppose now, that the last term is 2^ositive. 

Making x = in the first member, we obtain + U for the result; 
but by putting — (-£'+ 1) in place of x., we shall obtain a negor- 
live result, since the first term becomes negative by this sdb 
stitution ; hence, the equation has at least one real root com 
prehended between and — (^-f 1), which is negative, oi 
affected with a sign contrary to that of the last term. 



GHAP. XI. 1 CONSEQUENCES OF PRINCIPLES. 861 

Fourth. 

290. Every equation of an even degree^ which involves only real 
CO -efficients, and of which the last term is negative^ has at least two 
real roots, one positive and the other negative. % 

For, let — U he the last term ; making x := 0, there results 
— U. Now, substitute either ^+1, or — (^+ 1), -^ being 
the greatest co-efficient in the equation. As m is an even number, 
the first term x^ will remain positive; besides, by these substi- 
tutions, it becomes greater than the sum of all the others ; there- 
fore, the results obtained by these substitutions are both positivCj 
or affii^cted with a sign con%ary to that given by the hypothesis 
a: = ; hence, the equation has at least two real roots, one positive, 
and comprehended between and K-{- 1, the other negative, and 
comprehended between and — (A'-f 1) (277). 

Fifth. 

291* If an equation, involving only real co-efficients, contains imagi- 
nary roots, the number of such roots must he even. 

For, conceive that the first member has be^n divided by all the 
simple factors corresponding to the real roots; the co-efficients 
of the quotient will be real (Art. 24G) ; and the quotient must alsc 
he of an even degree ; for, if it was uneven, by placing it equal 
to zero, we should obtain an equation that would contain at least 
one real root (289) ; hence, the imaginary roots must enter 
by pairs. 

Remark. — There is a property of the above polynomial quotient 
which belongs exclusively to equations containing only imaginary 
roots ; viz., every such equation always remains positive for any 
real value suhstituted for x. 

For, by substituting for x, K -\-\, the greatest co-efficient 
plus 1, we coLild always obtain a positive result; hence, if the 
polynomial could become negative, it would follow that when 
placed equal to zero, there ^'Ould be at least one real root com- 



362 ELEMENTS OF ALGEBRA. [CHAP. XI. 

prehended between K -\- I and the number which would give a 
negative result {Art. 277). 

It also follows, that the last term of this polynomial must be 
positive^ otherwise x = would give a negative result. 

• Sixth. 

292. When the last term of an equation is positive, the number 
of its real positive roots is even ; and when it is negative, the 
number of such roots is uneven. 

For, first suppose that the last term is + U, or j^osiiive. Since 
by making x = 0, there will result + U, and by making x — K-{-l, 
the result will also be positive, it ^ibllows that and A" + 1 
give two results affected with the same sign, and consequently 
(Art. 279), the number of real roots, if any, comprehended be- 
tween them, is even. 

Yv^hen the last term is — U, then and K -\- \ give two 
results affected with contrary signs, and consequently, they com- 
prehend either a single root, or an odd number of them. 

The converse of this proposition is evidently true. 

Descartes' Rule. 

293. An equation of any degree whatever, cannot have a greater 
number of positive roots than there are variations in the signs of 
its terms, nor a greater number of negative roots than there are 
permanences" o/" these signs. 

A variation is a change of sign in passing along the terms. A 

ferwMnence is when two consecutive terms have the same sign. 

In the equation 

.r — a — 0, 

there is one variation, and one positive root, z =z a. 

And in the equation x -\- b ^=: 0, there is one permanence, and 
one negative root, x = — b. 

If these equations be multiplied together, member by member, 
there will resul!; an equation of the second degree, 
x^ — a 



+ b 



"' \ = 0. 



CRAF. XI. J DESCARTES' RULE. 863 

If a is less than b, the equation will be of the firso form 
(Art. 117); and if a ^ b, the equation will be of the second 
form ; that is, 

a <Cb gives x"^ -j- ^P^ ~ Q =^ ^i 
and a^'b " x^ — 2px — q z=z 0. 

In the first case, there is one permanence and one variation, 
and in the second, one variation and one permanence. Since 
in either form, one root is positive and one negative, it fol- 
lows that there are as many positive roots as there are 
variations, and as many negative roots as there are perma- 
nences. \ 

The proposition will evidently be demonstrated in a general 
manner, if it be shown that the multiplication of the first mem- 
ber of any equation by a factor x — a, corresponding to a posi- 
tive root, introduces at least one variation, and that the multi- 
plication by a factor a; -f «, corresponding to a negative root, 
introduces at least one permanence. ^ 

Take the equation, ' 

x'^ dc Ax"^-^ ± Bx-^-"^ ± Cx"^-^ ± , . . dt Tx±: U=iO, 
!n which the signs succeed each other in any manner whatever. 
By multiplying by x — a, we have 
^.m+i-j-^ x^^dcB x^-^±Q |a:"'-2dz . . . =h C^ 
— - a :=fAa zjpBa \ =F ^cl 

The co-efficients which form the first horizontal line of this 
product, are those of the given equation, taken with the same 
signs ; and the co-efficients of the second line are formed from 
those of the first, by multiplying by a, changing the signs, and 
advancing each one place to the right. 

Now, "sO long as each co-efficient in the upper line is greater 
than the corresponding one in the lower, it will determine the 
sign of the total co-efficient ; hence, in this case there will be, 
from the first term to that preceding the last, inclusively, the 
same variations and the same permanences as in the proposed 
equation ; but the last term rp Ua having a sign contrary to that 
which immediately precedes it, there must De one more varia- 
tion ihan in the proposed equation. 



^^c/J = ^- 



364 ELEMENTS OF ALGEBRA. [CHAP. XI. 

When a co-efficient in the lower line is affected with a sign 
contrary to the. one corresponding to it in the upper, and is 
also greater than this last, there is a change from a perma 
nence of sign to a variation ; for the sign of the term in whicn 
this happens, being the same as that of the inferior co-efficient, 
must be contrary to that of the preceding term, which has 
been supposed to be the same as that of the superior co-effi- 
cient. Hence, each time we descend from the upper to the 
lower Ime, in order to determine the sign, there is a variation 
which is not found in the proposed equation ; and if, after 
passing into the lower line, we continue in it throughout, we 
shall find for the remaining terms the same variations and the 
same permanences as in the given equation, since the co-efficients 
of this line are all affected with signs contrary to those of the 
primitive co-efficients. This supposition would therefore give us 
one variation for each positive root. But if we ascend from 
the lower to the upper line, there may be either a variation 
or a permanence. But even by supposing that this passage pro- 
duces permanences in all cases, since the last term qp Ua foiTins 
a part of the lower line, it will be necessary 'to go once "xiore 
fi'om the upper line to the lower, than from the lower to the 
upper. Hence, the new equation must have at least one more 
variation than the proposed ; and it will be the same for each 
positive root introduced into it. 

It may be demonstrated, in an analogous manner, that the 
multiplication of the first member hy a factor x -\- a^ correspond 
ing to a negative root, would introduce one permanence more. 
Hence, in any equation, the number of positive roots cannot be 
greater than the number of variations of signs, nor the number 
of negative roots gi eater than the number of permanences. 

Consequence. 

294. When the roots of an equation are all real, the number 
of positive roots is equal to the mimber of variations, ujid the num- 
ber of negative roots to the number of pe*-manences. 



CHAP. XI.J DESCARTES' RULE. "^ 365 

For, let m dm.ote the degree of the equation, n the nunjber 
of variations of the signs, p the number of permanences ; then, 

m =z n -\- p. 
Moreover, let n^ denote the number of positive roots, and p^ 
the number of negative roots, vre shall have 

m = ?^' + J9^ ; 
whence, n -\-p = n' -\- p^, or, n — n'z=p^—p. 

Now, we have just seen that n^ cannot be > n, nor can it be 
less, since p'' cannot be '^p ; therefore, we must have 
^ n^ z=: n, and p^ — p. 

Remark. — When an equation wants some of its terms, we can 
often discover the presence of imaginary roots, by means of the 
above rule. 

For example, take the equation 

x^ -\- px -{- q = {}^ 
p and q being essentially positive; introducing the term which 
is wanting, by affecting it with the co-efficient i ; it becomes 
x^ ± . x'^ -^ px -}- q =z 0. 
By considering only the superior sign, we should obtain only 
permanences, whereas the inferior sign gives two variations. This 
proves that the equation has some imaginary roots ; for, if they 
were all three real, it would be necessary, by virtue of the supe- 
rior sign, that they should be all negative, and, by virtue of the 
inferior sign, that two of them should be positive and one negOr 
tive, which are contradictory results. 

We can conclude nothing from an equation of the form 
x^ — px -\- q = ', 
for, introducing the term zh .x^, it becomes 

x^ ± . x"^ — px -{- q = 0, 
which contains one permanence and two variations, whether we 
take the superior or inferior sign. Therefore, this equation may 
have its three roots real, viz., two positive and one negative ; 
or, two of its roots may be imaginary and one negative, since 
its last term is positive (Art. 292). 



366 ELEMENTS OF ALGEBRA. LCHAP. XI 

Of the commensurahle Roots of Numerical Equatioi.s. 

295. Every equation in which th^ co-efficients are whole mini- 
bers, that of the first term being 1, will have whole numbera 
only for its commensurable roots. 

Tor, let there be. the equation 

^m _|_ p^m-\ ^ Q^m-2 J^ ^ ^ ^ J^ rpx -\- U = \ 

in which P. Q , . . T, U, are whole numbers, and suppose that 

a 

It were possible for one root to be an irreducible fraction — . 

Substituting this fraction for x, the equation becomes 

whence, multiplying both members by b^-"^, and transposing, 
J- = — Po^"^^ — QoJ^-'^h — ... — Tah^-"^ — TJh-^-^. 

But the second member of this equation is composed of 
the sum of entire numbers, while the first is essentially frac- 
tional, for a and h being prime with respect to each other, aJ^ 
and h will also be prime with respect to each other (Art. 95), 
and hence this equality cannot exist; for, an irreducible frac- 
tion cannot be equal to a whole number. Therefore, it is im- 
possible for any irreducible fraction to satisfy the equation. 

Now, it has been shown (Art. 262), that an equation con- 
taining rational, but fractional co-efficients, can be transformed 
into another in which the co-efficients are whole numbers, 
that of the first tevm being 1. Hence, the search for commensu- 
rahle roots, either entire or fractional, can always be reduced to 
that for entire roots. 

296. This being the case, take the general equation 

X^n ^ p^m-l 4. Qx^"^ . . . -{- Rx^ + Sx"^ ■\- Tx ^ TJ ^ 0, 

and let a denote any entire number, positive or negative, wliiclj 
will satisfy it. 

Since a is a oot, we shall have the equation 
o" -I- Fa^^-\- . . . + i2a3 4- AS'a2 + Ta -f U— - (1). 



CHAP. XI.] COMMENSURABLE ROOTS OF EQUATIONS. 367 

Now replace a By all the entire numbers, positive and negative, 
between 1 and the linait -\-L, and between — 1 and — L^^ : those 
which verify the above equality will be roots of the equation. 
But these trials being long and troublesome, we will deduce from 
equation (1), other conditions equivalent to this, and more easily 
applied. 

Transposing in equation (1) all the terms except the last, and 
dividing by a, we have, 

^ = - a'"-! - Pa'^2 _ _ ^ _jia^ _ Sa— T - - - (2). 



a 

Now, the second member of this equation is an entire number ; 

hence, — must be an entire number : therefore, the entire roots of 
a 

the equation are comprised among the divisors of the last term. 

Transposing — T' in equation (2), dividing by a, and making 

— -{- T=r, we have, 
a 

rjv 

— = - a'--2 - Pa^^ . . . —Ra — S - - - - (3), 
a ^ ^ 

T> 

The second member of this equation being entire, — , that is, 

the quotient of 

U 

~+r by a, 

ts an entire number. 

Transposing the term — S and dividing by a, we have, by 

supposing 

T 

a 
cf/ 
— = - a*^ - Pa^"^ — . . . —R... (4), 

The second member of this equation being entire. —, that is, 

the quotient of 

T 

— + >Sf by a, 

is an entire number. 



368 ELEMENTS OF ALGEBRA. [CHAP XI. 

By continuing to transpose the terms of the second member 
into the first, we shall, after m — 1 transformations, obtain an 
aquation of the form, 

a 

Then, transposing the term — P, dividing by a, and making 

O' P' P' 

^^ P = P\ we have — = — 1, or h 1 = 0. 

a . ' a a 

This equation, which results from the continued transforma^ 
tions of equation (1), expresses the last condition which it ii 
requis-ite for the entire number a to fulfil, in order that it may 
be known to be a root of the equation. 

297. From the preceding conditions we conclude that, when 
an entire number a, positive or negative, is a root of the given 
equation, the quotient of the last term, divided hij a, is an 
(mtire number. 

Adding to this quotient the co-efficient of x^, the sum will 
be exactly divisible by a. 

Adding the co-efficient of x"^ to this last quotient, and . again 
dividing by a, the new quotient must also be entire; and so on. 

Finally, adding the co-efficient of the second term, that is, of 
x^"-~^, to the preceding quotient, the quotient of this sum divided 
by a, must be equal to — \ \ hencB, the result of the addition of 
1, which is the co-efficient of x"^, to the preceding quotient, must 
be equal to 0. 

Every number which will satisfy these conditions will be a 
root, and those which do not satisfy them should be rejected. 

All the entire roots may be determined at the same time, 
by the folio wmg 

RULE, > 

After having determined all the divisors of the last term, write 
those which are comprehended between the limits -\- L and — \/^ 
upon the same horizontal line ; then underneath these divisors 'write 
iJie quotients of the last term by each of them. 



CHAP. XI.] 



COMMENSURABLE ROOTS. 



369 



Add the co-efficieiit of x^ to each of these quotients, and write 
the sums underneath the quotients which correspond to them. 
Then divide these sums by each of the divisors, and write the quo- 
iients underneath the corresponding sums, taking care to reject the 
fractional quotients and the divisors which produce them ; aiid 
so on. 

When there are terms wanting in the proposed equation, 
their co-efficients, which are to be regarded as equal to 0, must 
be taken into consideration. 

. * EXAMPLES. 



1. What are the entire roots of the equation, 
x^ — x"^ — 13a:2 + 16.2:— 48 = 0? 

A superior limit of the positive roots of this equation (Art.. 
284), is 13 + 1 = 14. The co-efficient 48 need not be con- 
sidered, since the last two terms can be put under the form 
16 (.r — 3) ; hence, when a; > 3, this part is essentially positive. 

A superior limit of the negative roots (Art. 286),- is 

-(l-fyiS), or -8. 

Therefore^ the divisors of the last term which may be roots, 
are L 2, 3, 4, 6, 8, 12; moreover, neither -f 1, nor — 1, will 
satisfy the equation, because the co-efficient —48 is itself greater 
than the sum of all the others : we should therefore try only 
the positive divisors from 2 to 12, and the negative divisors from 
— 2 to — 6 inclusively. 

By observing the rule given above, we have 



12, 


8, 


- 4, 


- 6, 


+ 12, 


4 10, 


f 1, 


.., 


-12, 


••> 


- 1, 


. 


- 3, 


••» 


.., 


••» 



6, 4, 


3, 


2, 


- 2, 


-3,-4,-6 


- 8, - 12, 


-16, 


-24 


+ 24, 


+ 16, + 12, + 8 


+ 8, + 4, 


0, 


- 8, 


+ 40, 


+ 32, + 28, + 24 


•., + 1, 


0, 


- 4 


-20, 


.., -^7,-4 


..; - 12, 


-'3, 


-17 


, -33, 


.., -20, -17 


.., - 3, 


-5 


.. 


.., 


., + 5, 


., - 4, 


.., 


.. 


.., 


.„ -f- 4, 


., - 1, 


.., 


•• 


.., 


.., - 1, 



24 



870 ' . ELEMENTS OF ALGEBRA. [CHAP. XI 

The jir&t line contains the divisors, the second contains the 
quotients arising from the division of the last term — 48, by 
each of the divisors. The third line contains these quotients, each 
augmented by the co-efficient -f- 16 ; and the fourth^ the quotients 
of these sums by each of the divisors ; this second condition 
excludes the divisors +8, +6, and — 3. 

The fifth contains the preceding line of quotients, each aug 
mented by the co-efficient — 13, and the sixth contains the quo 
tients of these sums by each of the divisors ; the third condition 
excludes the divisors 3, 2, — 2, and — 6. 

Finally, the seventh is the third line of quotients, each aug 
mented by the co-efficient — 1, and the eighth contains the quo- 
tients of these sums by each of the divisors. The divisors -f 4 
and — 4 are the only ones which give — 1 ; hence, -f- 4 and 
— 4 are the only entire roots of the equation. 

In fact, if we divide 

x^ — x^ — 13.1-2 -f 16a; — 48, 

by the product {x —A) {x + 4), or x"^ — 16, the quotient will 
be x"^ — iJ? + 3, which placed .equal to zero, gives 

JL^ 1 

~2 



. = ^±_/^^ 



liierefore, the four roots are- 

4,-4, l + l/^TT and 1-1-/^11. 

2. What are the entire roots of the equation 

ar* — 5a;3 + 25a; — 21 = ? 
S. What are the entire roots of the equation 

15j;5 _ 19a;* + 6a;3 + Ibx"^ - 19a; +^6 = ? 

4. WTiat are the entire roots of the equation 

9a;« -f- 30a;S -f 22ar* -|- lOa;^ -f llx"^ - 20a; -f- 4 = ? 



CHAP. XI.J Sturm's theorem. 371 

Sturm's Theorem. 

298. The object of this theorem is to explain a method of do- 
terniiiiing the number and places of the real roots of equations 
involving but one unknown quantity. » 

Let XrrO - - - - (1), 

represent an equation containing the single unknown quantity x ; 
X being a polynomikl of the rn^^ degree with respect to-ar, the 
co-efficients of which are all real. If this equation should have 
equal roots, they may be found and divided out as in Art. 2(>9, 
and tho reasoning be applied* to the equation which would result^ 
We will therefore suppose X = to have no equal roots. 

299. Let us denote the first derived polynomial of X by X„. 
and then apply to X and X^^ a process similar to that for find- 
ing their greatest common divisor, differing only in this respect, 
that instead of using the successive remainders as at first ob- 
tained, we change their signs, and take care also, in jjreparing for 
the division, neither to introduce nor reject any factor except a 
positive one. 

If we denote the several remainders, in order, after their signs 
have been changed, by X^, X^ . . . X^, which are read X second, 
X third, <Src., and denote the corresponding quotients by Q^, Q^ 
. . Qr-u ^^^ niay then form the equations 

X=X,Q,-X, .... (2). 



X„^„-X„+, )> - - - (3). 



Xr—<i Xr—\ Qr—\ X, 



Since by hypothesis, X = has no equal roots,, no eommoQ 
divisor can exist between X and X^ (Art. 267). The last re. 
imainder — Jr„ will therefore be different from zero, and inde- 
pendent of re. 



872 ELEMENTS OF ALGEBRA. [CHAP. XL 

EOO. Now, let us suppose that a number p has been substi 
toted for x in each of the expressions X, Xj, Xj . . . X^i ; 
and that the signs of the results, together with the sign of X,, 
are arranged in a line one after the other : also that another 
number q, greater than p, has been substituted for a:, and the 
egns of the results arranged in like manner. 

Then will the number of variations in the signs of the first 
arrangement, diminished hy the number of variations in those of 
the second, denote the exact number of real roots comprised be- 
tween p and q. ^ 

301. The demonstration of this truth mainly depends upon 
tiie three following properties of the expressions X, Xj . . X„, &c. 

1. If any number be: substituted for x in these expressions, it is 
impossible that any two consecutive ones can become zero at the 
same time. 

For, let X^i, X^, X„+i, be any three consecutive expressions, 
llien among equations (3), we shall find 

X„-,=:X„^„-X„+,....(4), 

from which it appears that, if X^j and X„ should both become 
for a value of x, X^^-j would be for the same value ; and 
since the equation which follows (4) must be 

X„ =: X^+i $„_{.i — X„4.2, 

we shall have X„4.2 = for the same value, and so on until 
we should find X, — 0, which cannot be ; hence, X^j and X^ 
cannot both becomxC for the same value of x. 

II. By an examination of equation (4), we see that if X, be- 
comes for a value of a;, X^, and X,+i must have contrary 
i^gns ; that is. 

If any one of the expressions is reduced to by the substi- 
tution of a value for x, the ptreceding and following ones will 
have contrary signs for the same value. 



CHAP. XI. j Sturm's theorem. 373 

111. Let us substitute a -j- u for x in the expressions X and 
Xi, and designate by U and Ux what tney respectively become 
under this supposition. Then (Art. 264), we have 

U =A -\-A'u +A'''^ + &c. 1 
U,=:A + A\u 4- A'\-j + &c. J 

in which A, A^, A^^, &c., are the results obtained by the sub 
stitution of a for x, in X and its derived polynomials ; and 
^1, ^^1, &;c., are similar results derived from X-^. If, now, a be 
a root df the proposed equation X == 0, then A =^ 0, and since 
^' and Ai are each derived from Xj, by the substitution of 
a for a:, we have A'' =. A^, and equations (5) become 

U=A'u + A"- + ^. . . . (6). 



Ux:=A' + A\u + &c. 

Now, the arbitrary quantity u may be taken so small that 
the signs of the values of U and U^ will depend upon the 
signs of their first terms (Art. 276) ; that is, they will be alike 
when u is positive, or when a -\- u i^ substituted for x^ and ua 
like when u is negative or when a — u is substituted for x. 
Hence, 

If a number insensibly less than one of the real roots of 

X = 5e substituted for x in X and X^, the results will have 

contrarij signs ; arid if a number insensibly greater than this root 
be substituted^ the results will have the same sign. 

302. Now, let any number as k, algebraically less, that is, 
nearer equal to — oo, than any of the real roots of the seveial 
equations 

X=0, Xi = . . . X,_, = 0, 

be substituted for x in the expressions X, Xj, Xj, &;c., and the 
signs of the several results arranged in order ; then, let x bo 
increased by insensible degrees, until it becomes equal to h 
the least of all the roots of the equations. As there is no 



S74 ELEMENTS OF ALGEBRA. [CHAP. XI. 

root of either of the equations between k and h, none of the 
sign's can change while x is less than h (Art. 277), and the 
number of variations and permanences in the several sets of 
results, will remain the same as in those obtained bj the first 
substitution. 

When X becomes equal to 7i, o)ie or moi-e of the expressions 
X, X, &;c., will reduce to 0. Suppose X„ becomes 0. Then, 
as by the first and second properties above explained, neither 
X„_i nor X„+i can become at the same time, but must have 
contrary signs, it follows that in passing from one to the other 
(omitting JC„ =: 0), there will be one and oiilt/ one variation ; 
and since their signs have not changed, one must be the same 
as, and the other contrary to, that of X„, both before and after 
it becomes ; hence, in passing over the three, either just before 
X„ becomes or just after, there is one and onli/ one variation. 
Therefore, the reduction of X„ to neither increases nor di- 
minishes the number of variations ; and this will evidently be 
the case, although several of the expressions JTi, Xj, &;c., should 
become at the same time. 

If X =: h should reduce X to 0, then h is the least real root 
of the proposed equation, which root w^e denote by a ; and 
since by the third property, just before x becomes equal to a, 
the signs of X and X^ are contrary, giving a variation, and just 
after passing it (before x becomes equal to a root of X^ = 0), 
the signs are the same, giving a permanence instead, it follows 
that in passing this root a variation is lost. 

In the same way, increasing x by insensible degrees from 
X z= a -}- u until we reach the root of X = 6 next in order, it 
is plain that no variation will be lost or gained in passing any 
of the roots of the other equations, but that in passing this 
root, for the same reason as before, another variation will be 
lost, and so on for each real root between k and the number 
last substituted, as g, a variation will be lost until x has been 
iBCi eased beyond the greatest real root, when no more can he 
iost or gained. Hence, the excess of the number of variations 



CHAP. xT.j s^^urm's theorem. 375 

obtained by the substitution of k over those obtained by the 
substitution of g, will be equal to the number of real roots 
comprised between k and g. 

It is evident that the same course of reasoliing will apply 
when we commence with any number 2^1 whether less than all 
the lootJB or not, and gradually increase x until it equals any 
other number q. The fact enunciated in Art. 299 is therefore 
established. 

303. In seeking the number of roots comprised between p and gj 
should either p or q reduce any of the expressions Xi, Xj, &;c., 
to 0, tl^ result Will not be affected by their omission, since 
the number of variations will be the same.. 

■ Should p reduce X to 0, then /> is a root, but not one of those 
sought ; and as the substitution o^ p -\- u will give X and JT, 
the same sign, the number of variations to be counted will not 
be affected by the omission of X == 0. 

Should q reduce X to 0, then q is also a root, but not one 
of those sought ; and as the substitution oi q — u will give X 
and Xj contrary signs, one variation must be counted in passing 
from X to Xj. 

304» If in the application of the preceding principles, we ob- 
serve that any one of the expressions Xj, Xj . . . &c., X» for 
instance, will preserve the same sign for all values of x in 
passing from p to q^ inclusively, it will be unnecessary to use 
the succeeding expressions, or even to deduce them. For, as 
X« preserves the §ame sign during the successive substitutions, 
it is plain that the same number of variations will be lost 
among the expressions X, Xj, &c. . . . ending with X„ as among 
all including X,. Whenever then, in the course of the division, 
it is found tha^t by placing any of the remainders equal to 0, 
an equation is obtained with imaginary roots only (Art. 291), 
it will be useless to obtain any of the succeeding remainders. 
This principle will be found very useful in the solution of 
numerical examples. 



376 ELEMENTS OF ALGEBRA. [CHAP. XI 

305i As all the real roots of the proposed equation are neces- 
sarily included between — oo and + oo, we may, by ascertain- 
ing the number of variations lost by the substitution of these, 
in succession, in the expressions X, X{ . . . X„, . . &c., readily 
determine the total number of such roots. It should be ob- 
served, that it will be only necessary to make these substitu- 
tions in the first terms of each of the expressions, as in this 
case the sign of the term -s^-ill determine that of the entire ex- 
pression (Art. 282). 

Having found the number of real roots, if we subtract this 
number from the highest exponent of the unknown quantity, the 
remainder will be the number of imaginary roots (Art. 248). 

306 1 Having thus obtained the total number of real roots, 
we may ascertain their places by substituting for x, in succes- 
sion, the values 0, 1, 2, 3, &c., until we find an entire num- 
ber which gives the same number of variations as -f oo. This 
will be the smallest superior limit of the positive roots in entire 
numbers. 

Then substitute — 1, — 2, &c., until a negative number is 
obtained which gives the same number of variations as — oo. 
This will be, numerically, the least superior limit of the 
negative roots in entire numbers. Now, by commencing with 
this limit and observing the number of variations lost in passing 
from each number to the next in order, we shall discover how 
many roots are included between each two of the consecutive 
numbers used, and thus, of course, know the entire part of eacik 
root. The decimal part may then be sought by some of thi» 
known methods of approximation. 

EXAMPLES. 

1. Let bx^ — 6x—l = = X. 

The first derived polynomial (Art. 264), is 

24^2 _ e^ 



CHAP. XI.] Sturm's theorem. 877 

and since we may omit the positive factor 6, without affecting 
tlie sign, we may write 

Dividing X by Xj, we obtain for the first remainder, — 4a:--l. 
Changing its sign, we have 

4^ 4- 1 = Xj. 

Multiplying Xj by the positive number 4, and then dividing 
by X2, we obtain the second remainder —3; and by changing 
its sign 

■ +S = X,. 

The expressions to be be used are then 

X=zSx^-6x- 1, X, = 4:^2 - 1, X, = 4x^-1, X, = -\- 3. 

Substituting — 00 and then + oc), we obtain the two following 
arrangements of signs : 

— + — -\- 3 variations^ 

+ 4- + + « 

there are then three real roots. 

If, now, in the same expressions we substitute and + 1, 
and then and — 1, for a:, we shall obtain the three following 
arrangements : 

For X = -\- \ -\ — |-H — h variations. 

« x= 1-^-1 « 

" x= -\ 1 h 3 " 

As x =: -\- \ gives the same number of variations as -J- 00, 
and X z=z — \ gives the same as — 00, 4- 1 and — 1 are the 
smallest limits in entire numbers. In passing from — 1 to 0, 
twc variations are lost, and in passing from to +1, om 
variation is lost ; hence, there are two negative roots between 
— 1 and 0, and one positive root between and -f 1. 

2. Let 2a:* — ISa:^ + \0x - 19 = 0. 



878 ELEMENTS OF ALGEBRA. [CHAP. XI 

If we deduce X, Xj, and X, we have the three expressions 
X = 2x^- ISx^ 4- 10a; — 19, 
X, = Ax^ - \^x + 5, 
X^ = 13;r2 _. 15^ _|_ 38, 

If we place X.^ = 0, we shall find that both of the roots of 
the resulting equation are imaginary ; hence, X^ will be positive 
for all values of x (Art. 290). It is then useless to seek for 
Xg and X4. 

By the substitution of — oc and -{- cx> m X^ X^, and X^, we 
obtain for the first, two variations, and for the second, none; 
hence, there are two real and two imaginary roots in the 
proposed equation. 

3. Let x^ — bx^ + 8:c — 1 = 0. 

4. X* — x^ — 3a:2 + a;2 — a; — 3 = 0. 

5. . a;5 - 2^:3 -h 1 rzz 0. 
Discuss each of the above equations. 

307t In the preceding discussions we have supposed the 
equations to be given, and from the relations existing between 
the co-efficients of the different powers of the unknown quan- 
tity, have determined the number and places of the real roots; 
and, consequently, the number of imaginary roots. 

In the equation of the second degree, we pointed out the 
relations which exist between the co-efficients of the different 
powers of the unknown quantity when the roots are real, 
and when they are imaginary (Art. 116). 

Let us see if we can indicate corresponding relations among 
the co-efficients of an equation of the third degree. 

Let us take the equation, 

x^ + Fx^-^ Qx-{- U=0, 

and by causing the second term to disappear (Art. 263), it 
will take the form, 

x^ +px i- q = 0, 



CHAP. XI caedan's kule. 379 

Hence we have 

X = x^ -\- px -{- g, / 

X, = Sx'^+p, 
Xg = — 2px — oq, 

X^= - 4^3 _ 27^2, 

In order that all the roots be real, the substitution of ao for 
X in the above expressions must give' three permanences; and 
'the substitution of — oo for x must give three variations. But 
the first supposition can only give three permanences when 

_4p3_ 27^2^0; 

hat is, a positive quantity, a condition which requires that p be 
negative. 

If, then, p be negative, we have, for a: = oo, • 

_4p3 _ 27(^2 \, . ^}^jj|3 ig^ positive : 
or, Ap^ + 27^2 <- . that is, negative : 

(72 r)3 

hence, T "I" 97 '^ ^' "^^^^^ requires that p be 

negative, and that "^^^ > "j ; conditions which indicate tha.t the 
roots are all real. 

GardaTbS Rule for Solving Cubic Equations. 

308. First, free the equation of its second term and we 
have the form, 

x^ -\- px -\- q =1 (1). 

Take x =y -\- z\ 

then a:3 = ?/3 + 03 4- g^^; (y ■\- z)-, 

or, ])y transposing, and substituting x for y -\- 0, we have 
x^-Zyz.x-{y^ + z^) = - - - .(2); 
and by comparing this with equation (1), we have 
— Syz =: p ; and y^ 4 z^ = — q. 



380 ELEMENTS OF ALaSBRA, 

From the 1st. we have 



[CHAP. XL 



i^=z - 



pi 



27y3' 



which, being substituted in the second, gives 
3 P^ 

cr clearing of fractions, and reducing 



2/6 + qi/ 



27' 



Solving this trinomial equation (Art. 124), we have 



^=\/-I+\/t+6' 

and the corresponding value of z is 



V^fV 



q^ 'P2_ 
4 "^ 27* 



But since ar = ?/ + 0, we have 



This is called Cardaii's formula. 

By examining the above formula, it will be seen, tha.. it is 
>^iapplicable to the case, when the quantity 

4 "^ 27' 

under the radical of the second degree, is negative ; and hence 
is applicable only to the case where two of the roots are imag 
inary (Art. 307). 

Having found the real root, divide both members of the giveL 
equation by the unknown quantity, minus this root (Art. 247) ; 
the result will be an equation of the second degree, the roots 
of which may be readily found. 



CHAP. XI.J hokner's method. ZSl 



EXAMPLES. 



1. What are the roots of the equation 
a:3 __6;c2^ 10a; = 8'? 



Ans. 4, 1 +V^^ i-^-h 

2. What are the roots of the equation 

x^ .— 9x^ -\- 2Sx =z SO "i 

Ans. 3, 3+.^^=T, 3-y-l. 

3. What are the roots of the equation 

x3 — 7x^ -\-Ux =201 

Ans. 5, 1+y^^^, l-^-'S. 

Preliminaries to Horner''s Method. 

309. Before applying the method of Horner to, the solution 
of numerical equations, it will be necessary to explain, 

1st. A modification of the method of multiplication, called 
the method by Detached Co-efficients : 

2d. A modification of the method of division, called, also, the 
method by Detached Co-efficients : 

3d. A second modification of the method of division, called 
Synthetical Division : and, 

4th. The application of these methods of Division in the 
Transformation of Equations. 

Multiplication hy Detached Co-efficients, 

310. When the multiplicand and multiplier are both homo- 
geneous (Art. 26), and contain but two letters, if each be ar- 
ranged according to the same letter, the literal part, in the 
several terms of the product, may be written immediately, since 
the exponent of the leading letter will go on decreasing from 
left to right by a constant number, and the sum of the exponents 
of both letters will be the same, in each of the terms. 



ELEMENTS OF ALGEBRA, ICHAP. XI. 



EXAMPLES. 



1 Let it be required to multiply 

x"^ + x^y + xy^ -{- y^ hj x — y. 
Since a;^ X a; = a^, the terms of the product will be of tlie 
4th degree, and since the exponents of x decrease by 1, and 
those of y increase by 1, we may wi'ite the literal parts thus, 
rr*, x^y, xhf, xy^,, y*. 
In regard to the co- efficients, we have, : 

Co-efficients of multiplicand, ---1 + 1 + 1 + 1. 
" " multiplier, - 1 — 1 

1+1+1+1 
_ 1 _1 _1 _1 



co-efficients of the product, - - 1 + + + 0—1; 

aL.d writing these co-efficients before the literal parts to which 
they belong, we have 

.r* + . a;3?/ + . x'^y^ + . xy^ — y^ = x^ — y\ 

2. Multiply 2a3 — Sab^ + 56^ by 2a'^ - 552. 

In this example, the term a^b in the multiplicand, and ab in 
the multiplier, are both wanting ; that is, their co-efficieuts are 
0. Supplying these co-efficients, and we have, 

Co-efficients of multiplicand, - 2 + — 3 + 5 
" "* multiplier, - - 2 + — 5 











4 + 


-6 + 10 






-efficients of the 


product, - 




-10- 0+15- 


-25 


CO 


- 4 + 


-16+10 + 15- 


-25. 


H 


enee, the 


product is, 4a^ - 


- 16a352 


+ 10a263+15a6*- 


-25 


8. 


Multiply 


x^- 


■ Sx^ + Sx - 


1 by 


a;2_2rc+l. 




4. 


Multiply 


y'- 


^ ^ 2 

■ya + — a^ 


by ?/2 


1 2 

+ ya^ • X " • 





Remark. — The method by detached co-efficients is also appli- 
cable to the case, in which the multiplicand and n^ltiplier con- 
tain but a single letter. The terms whose co-efficients are zero 
must be supplied, when wanting, as in the previous example?. 



CHAP. XI.' DETACHED CO-EFFICIENTS. 383 

EXAMPLES. 

1. What is the product of a* + Sa^ -f 1 by a^ — SI 
%. What is the product of Z»2 — 1 by 6 4- 2 ? 

Division hy Detached Co-efficients. 

31 !• When the dividend and divisor are both homogeneous 
and contain but two letters, the division may be performed by 
means of detached co-efficients, in the following manner : 

1. Arrange the terms of the dividend and divisor according 
to a common letter. 

2. Subtract the highest exponent of the leading letter of the divi- 
sor from the exponent of the leading letter of the dividend, and 
the remainder will be the exponent of the leading letter of 
the quotient. • ^ • 

S. The exponents of the letters in the other terms follow 
the same law of increase or decrease as the exponents in the 
corresponding terms of the dividend. 

4. Write down for division the co-efficients of the different 
terms of the dividend and divisor, with their respective signs, 
supplying the deiicieiicy of the absent terms with zeros. 

5. Then divide. the co-efficients of the dividend by those of 
the divisor, after the manner of algebraic division, and prefix the 
several quotients to their corresponding literal parts. 

EXAMPLES, 



2. Divide 


8a5-. 


- 4a4a: ~ 


-2a^x^-{-3?x^ by 


4a2 - «». 


Tlie 


literal 


part 


will be 










a3, 


a^x, ax^^ x^ ; 




and fcr the 


numerical co- 


efficients, 










8-4- 


-2-fl 4 + 0- 


IJ 








8 + 0- 


-2 2-1 










-4 


+ 1 










-4 


+ 1 



















884 ELEMENTS OF ALGEBRA. LCHAF. XI, 

hence, the true quotient is 2a^ — a^x-, the co-efficients after — 1, 
being each equal .to zero. 

3. Divide x* — Sax^ — Sa^x"^ + I8a^x — 8a* by x"^ — 2ax —2a\ 

4. Divide 10a* — 27a3x + S4:a^x^ — I8ax^ — Sx^ by 2a^ 
— Sax 4- 4:X^. 

Remark. — The method by detached co-efficients is also appli- 
cable to all cases in which the dividend and divisor contain but 
a single letter. The terms whose co-efficients are zero, must be 
supplied, when wanting, as in the previous examples. 

EXAMPLES. 

I. Let it be required to divide 

Qa"^ — 96 by 3a — 6. 
The dividend, in this example, may be written under t\e form, 

6a* + . a3 + . a^ -f . a — 96a0. 
Dividing a* by a, we have a^ for the literal part of the 
first term of the. quotient ; hence, the form of the quotient is 
a^, a^, a, a°. 
For the CO- efficients, we have, 

6-f0-f04-0-96||3-6, 

6—12 2-1-4 + 8 + 16 quotient ; 



hence, the true quotient is, 

2a3 + 4a2 + 8a + 16. 

Synthetical Division. 

312. In the common method of division, each term of the 
divisor is multiplied by the first term of the quotient, and the 
products subtracted from the dividend; but the subtractions are 
performed by first changing the sign of each product, and then 
adding. If, therefore, the signs of the divisor were first changed, 
we should obtain the same result by adding the products, instead 
of subtracting as before, and the same for any subsequent oper- 
ation. 



CHAF. Xt.J SYNTHETICAL DIVISION . 885 

By this process, the second dividend would be the same as 
by the common method. But since the second term of the quo. 
tient is found by dividing the first term of the second dividend 
by the first term of the divisor ; and since the sign of the latter 
has been changed, it follows, that the sign of the second term 
of the quotient will also be changed. 

To avoid this change of sign, the sign of the first term of the 
divisor is left unchanged, and the products of all the terms of 
the quotient by the first term of the divisor, are omitted; be 
cause, in the usual method, the first term.', in each successive 
dividend are cancelled by these products. 

Having made the first terra of the divisor 1 before commeno 
ing the operation, and omitting these several products, the co-effi- 
cient of the first term of any dividend will be t*ie co-efficient of the 
succeeding term of the quotient. Hence, the co-efficients in the 
qu )tient are, respectively, the co-efficients of the first terms of 
thi; successive dividends. 

The operation, thus simplified, may be fi^rther abridged by 
omitting the successive additions, except so much only as may 
be necessary to show the first term of each lividend ; and also, 
by writing the products of the several terms of the quotient by 
the modified divisor, diagonally, instead of horizontally, the first 
product falling under the second term of the dividend. 

Hence, the following 

RULE. 

I. Divide the (divisor and dividend hy the co-efficient of the first 
term of the divisor^ 'when that co-efficient is not 1. 

II. Write, in a horizontal line, the co-efficients of the dividend, 
with their proper signs, and -place the coefficients of the divisor^ 
with all their signs changed, except the first, on the right. 

III. Divide as -in the method hy detached co efficients, except that 
no term of the quotient is tnultiplied by the first term of the divi- 
sor, and that all the products are written diagonally to the right, 
under the terms of the dividend to which they correspond. 

25 



ELEMENTS OF ALGEBRA. [(5hAP. XI, 

rV. The first term of the quotiont is the same as that of the 
dividend ; the second term is the sum, of the numbers in the second 
column ; the third term^ the sum of the numbers in third column^ 
and so on, to the right. 

V. When the division can be exactly made, columns will be found 
at the right^ whose sums will be zero: when th§ division is not 
exact, continue the operation until a sufficient degree of approxi- 
mation ^tS^attained. Having found the co-efficients^ annex to them 
the literal parts, 

EXAMPLES. 

1. Divide 
a« - ba^x + lOa^x'^ — lOa^x^ + 5a^* — x^ bj a^ — 2aa: + x*. 

1-5 + 10-10 + 5-1 II 1 + 2-1 



2 - 6 + 6-2 
- 1 + 3-3 + 1 



3 + 3 



1 _ 3 4- 3 _ 1 0. 

Hence, the quotient is 

a3 — Za'^x + 3aa;2 — x\ 

Eemark. — The first term of the divisor being always 1, need 
fiot be written. The first term of the quotient is the same as 
that of the dividend. 

2. Divide 

««— 5a;5+15a;*— 24a;3+27a;2— 13a:+5 by x^-2x^^4:x'^—2x^\, 

1 _. 5 -I- 15 _ 24 + 27 - 13 + 5 | | 1 + 2-4 + 2- 1 
+ 2-6 + 10 .1^3^5 

-4 + 12-20 

+ 2 - 6+10 

-1+3-5 



1-3+5 00. 
Hence, the quotient is x"^ ~~ Zx •\- 5. 
3. Divide 

.«« + 2a46 + ZaW — ^263 _ 2a5* - 3^5 by a* + 2ah + 36». 
Ans. a3 + . a26 + . a52 — 63 _ a3 _ ja; 



CHAP XI. J SYNTHETICAL DIVISION. ' 387 

4. Divide 1 — x hj 1 + x. Ans. I — 2x -f- 2.r» - 2x^+ &o. 

5. Divide 1 by 1 — x. Ans. I + x -{- x'^ + x^ + &;c. 

0. Divide x"^ — y'' by x — y. 

Ans. x^ -\- x^y -{- x^y^ ■\- x'^y^ -\- x^y^ + xy'^ + y*. 

7. Divide a^ — 3a*a;2 .+ Za^x^ — x^ by a^ — Za?x + Saa;^ — x\ 

Ans. a^ 4- Sa^a; + 3aa;^' -f a;^''. 

313. To transform an equation into another whose roots shall be 
ike roots of the -proposed equation, increased or diminished hy a given 
quantity. 

A method of solving this problem has already been explained 
(Art. 264) ; but the process is tedious. We shall now explain 
a more simple method of finding the transformed equation. 

Let it be required to transform the equation 

ax"^ + Px"^^ + Qx"^-"^ .... Tx+ U= 

into another whose roots shall be less than the roots of this 
equation by r. 

If we write y -\- r for x, and develop, and arrange the terms 
with reference to y, we shall have 

^ym ^ p^ym-\ _|. Q^ym-2 , . , , -i^ Ty + U' =0 - - - (1)l 

But since y — x — r^ equation (1), may take the form 

a(x—rY-{-P\x—rY-^-\-Q\x—rY-'^ T{x—r)-{- 1^=0 (2), 

which, when developed, must be identical with the given equa?- 
tion. For, since y -\- r was substituted for x in the proposed 
equation, and then x — r for y in the transformed equation, we 
must necessarily have returned to the given equation. Hence, 
we have 

a[x — r)"^ + F'(x — r^-^ + Q'(x — ?-)'»-2 ... T{x — r)-^V 

= ax"" -f Pa.-'^i + Qx"^^ . . . Tx -{-U=0. 

If now we divide the first member by x — r, the quotient 
will be 

a{x — r)*"-! + P\x — r)'«-2 -f Q'{x — r)»*-3 . . » 3% 

and the remainder U\ 



SS8 ELEMENTS OF ALGEBKA. ICHAP. XI 

But since the second member is identical w th the first, the 
rerv same quotient and the same remainder would arise, if the 
second member were divided by x — r i hence. 

If the first member of the given equation be divided by the unknown 
quantity viinus the number which expresses the difference between th 
"ootSf the remainder will be the absolute term of the transformed equation. 

Again, if we divide the quotient thus obtained : viz., 
a{x — r)*"-! -}- F'(x — r)^"^ + Q'{x — r)"^^ . , . T 
by rr — r, the remainder will be T\ the co-efficient of the term 
last but one of the transformed equation ; and a similar result 
would be obtained by again dividing the resulting quotient 
by X — r. Hence, by successive divisions of the poly- 
nomial in the first member of the given equation and the quo- 
tients which result, by a: — r. we shall obtain all the co-eflicients 
of the transformed equation, in an inverse order. 

Remark. — When there is an absent term in the given equation. 
Its place miist be supplied by a 0. 

EXAMPLES. 

Transform the equation 

5a;* — Ylx^ + 3a;2 -f 4:i; — 5 = 
into anot) .^r whose roots shall each be less th^n those of the gives 
^uatJoT? »y 2. 

First Operation. 

5rr* - 12:^3 + 3a;2 -i- 4a; - 5 | | a; — 2 

5^-* - 1 0a:3 5.r5 - 2.r2 - a; ^- 2 



- 2a:3-f3a;2 






.^ 2x^-i-4x^ 






- x^ -\- ix 




~ x^ -^ 2x 






2x- 


-5 




2x~ 


-4 



1 1st remainder 



CHAP. XI.I SYNTHETICAL DIVISION. 380 

J 

Second Operation. 

5a;^ — 2a;2 - a? 4- 2 \ x —2 

5a;3 ~ \0x^ 5x^ -\-Sx-\-15 



Sx^- 


X 




8a:2- 


16x 






15x + 


2 




15a; — 


30 



32 2d remainder. 
Third Operation. Fourth Operation. 

— 2 5x+ 18||a; -2 



5ar2 + 8:^+15 
5a;2 — 10a; 



5a; +18 5a;-10i 



"Rh 



18a; + 15 28 4th remaindeF, 

18a; - 36 



51 3d remainder. 

Therefore, the transformed equation is 

5y* + 28z/3 ^ 51^2 + 32y — 1 = 0. 

This laborious operation can be avoided by the synthetical 
method of division (Art. 312). 

Taking the same example, and recollecting that in the syn- 
thetical method, the first term of the divisor not being used, may 
be omitted, and that the first term of the quotient, by which 
the modified divisor is to be multiplied for the first term of the 
product, is always the first term of the dividend ; the whole of 
the worlc may be thus arranged : 



CP=:~ 1 



5-12 


+ 3 


+ 4 


-5|^ 


10 


-4 


-2 


4 


- 2 


-1 


2 


-1 


10 


16 


30 




8 


15 


32 


.-. ^.-=32 


10 


86 


.-. e' 




18 


51 


= 51 


10 








28 


.'.P' 


= 28; 





890 ELEMENTS OF ALGEBRA. lCHAP. XI. 

for it is plain that the first remainder will fall under the abso- 
lute term, the second under the term next to the left, and so 
on. Hence, the transformed equation is • 

5y* + 28z/3 ^ 51^2 + 32y - 1 = 0. 

2. Find the equation whose roots are less by 1,7 than thos« 
of the equation 

^.3 _ 2a;2 + 3a: — 4 = 0. 

First, find an equation whose roots are less by 1, 
1-2 +3 -4[|1_ 
1 -1 2 



-1 2-2 

1 



2 

1 



1 

We have thus found the co-efficients of the terms of an equa- 
tion whose roots are less by 1 than those of the given equation ; 
the equation is 

a;3 -f a;2 -f 2a; — 2 = ; 
and now by finding a new equation whose roots are less than 
those of the last by .7, we shall have the required equation : thus, 

1 + 1 +2 -2||-^ 

.7 1.19 2^233 



1.7 . 3.19 .233 

.7 1.68" 



2.4 4.87 

.7 



3.1 

henc(^, the required equation is 

y3 + 3.1/- -f 4.87^ + .233 = 0. 

This latter operation can be continued from the former, with- 
out arranging the co-efficients anew. The operations have been 
explained separately, merely to indicate the several steps in the 



CHAP. ^T.] SYNTHEIICAL DIVISION. 391 

trausforination and to point out the equations, at each step 
resulting from the successive diminution of the roots. Com- 
bining the two operations, we have the following arrangement: 

1 -2 -h3 -4(1.7; or, 1-2 +3 -4(1.7 

I -1 2 1.7 - .51 4.233 



-1 


2 


-2 


1 





2.233 





2 


.233 


1 


1.19 




1.7 . 


3.19. 




.7 


1.68 





.3 2.49 .233 

1.7 2.38 



1.4 4.87 

1.7 

3J 



. 2.4 4.87 • 
.7 

ST 

We see, by comparison, that the above results are the same 
as those obtained by the preceding operations, 

3. Find the equation whose roots shall be less by 1 than 
the roots of 

a;3_7a; + 7 = 0. 

Ans. y3 _^ 3y2 — 4y -f 1 = o. 

4. Find the equation whose roots shall be less by 3 than 
the roots of the equation 

a;4 _ 3^.3 _ 15^2 4_ 49^ _ 12 :=^ 0. 

Ans, y^ + 9y3 + 12z/2 — My z= 0. 

5. Find the equation whose roots shall be less by 10 than 
the roots of the equation 

x^ + 22;3 -h 3a;2 + 4a; — 12340 = 0. 

Ans. 2/4 -f 42y3 -l. 663v2 -f 4664y = 0. 

6. Find the equation whose roots shall be less by 2 tliaa 
the roots of the equation 

x^ + 2a;3 — iSx^ ~ Idx — 0. 

Ans. ys r. 10^4 -I- ^2y- 4- 86y2 _{_ 70y + 4 = 0, 



392 ELEMENTS OF ALGEBRA. [CHAP. XI 

Hmmer^s Method of approxrnating to the Real Roots of 
Numerical Equations. 

314» The method of approximatmg to the roots of a numeri 
uaJ equation of any degree, discovered by the English inathe* 
matician W. G. Horner, Esq., of Bath, is a process of very 
remarkable simplicity and elegance. 

The process consists, simply, in a succession of transforma 
lions of one equation to another, each transformed equation, as 
it arises, having its roots equal to the difference between 
the true value of the roots of the given equation, and 
the part of the root expressed by the figures already 
foirnd. Such figures of the root are called the initial fgures. 
Let 

V= x"^ H- P.T'"-i -i- ^^•'"-^ . . . , -\-Tx-{- U=i - - - (1) 

be any equation, and let us suppose that we have foun ' a 
part of one of the roots, which we will denote by m, and de> 
aote the remaining part of the root by r. 

Let us now transform the given equation into another, w .ose 
roots shall be less by m, and we have (Art. 313), 

V = ?•'" + PV^i + Q'r^"^ . . . . + T'r + U' = ■ (2). 

Now, when r is a very small fraction, all the terms of tho 

second member, except the last two, may be neglected, and the 

first figure, in the value of r, may be found from the equaliou 

U' U' 

Tr + C7' r= ; giving - r == — ; or r = - - ; hence, 

The first figure of r is the first figure of the quotient obtained by 
dividing the absolute term of the transforrned equation by the penulti- 
mate co-efficient. 

If, now, we transform equation (2) into another, whose roots 
shall be less than those of the previous equation by the first 
figure of r, and designate the remaining part by s, we shall 
have, 

W = s'" + P^'s"* -1 + Q^'s'^-'^ .... J^ T's + W =. 0, 



CHAP. XI. I Horner's method. 393 

the roots of which will be. less than those of the given eqiia- 
tion by m + the first figure of r. The first figure in the value 
of s is found from the equation, 

T"s + U" = (i, giving s=^. 

We may thus continue the transformations at pleasure, and 
each one will evolve a new figure of the root. Hence, to find 
the roots of numerical equations. 

1. Find the number and places of the real roots by Sturms' 
theorem^ and set the negative roots aside. 

[I. Txansfonn the given equation into another ivhose roots shall 
he less than those of the given equation, by the initial figure or 
figures already found: then, by Sturms' theorem^ find the places 
of the roots of this neio equation, and the first figure of each will 
be the first decimal place in each of the required roots. 

III. Transform the equation again so that the roots shall be less 
than those of the given equation^ and divide the absolute term of 
the transformed equation by the penultimate co-efficient.^ which is 
called the trial divisor^ and the first figure of the quotient will be 
the next figure of the root. 

IV. Transform the last equation into another whose roots shall 
he less than those of the previous equation by the figure last found., 
and proceed in a similar manrier until the root be found to the 
required degree of accuracy. 

Remark I. — This method is one of approximation, and it may- 
happen that the rejection of the terms preceding the penultimate 
term will affect the quotient figure of the root To avoid Ihia 
source of error, find the first decimal places of vhe root, also, 
by the theorem of Sturm, as in example 4, page 399, and when 
the results coincide for two consecutive places of decimals, those 
subsequently obtained by the divisors may be relied on. 

Remark II. — When ^he decimal portion of a negative root is 
to be found, first transform the given equation into a'r.other by 
changing the signs of the alternate terms (Art. 280), and then 
find the decimal part of the corresponding positive root of 
this new equation. 



394 



ELEM^ENTS OF ALGEBRA. 



[CHAP. XI. 



no variation, 
three Yariations 



III. When several decimal places are found in the root, the 
operation may be. shortened according to the method of con- 
tractions indicated in the examples. 

314i Let us now work one example in full. Let us take the 
equation of the thii'd degree, 

x^ — 7x-\-7 = 0. • 

By Sturm's rule, we have the functions (Art. 299), 
X == .r3 _ 7^; + 7 
X, = 3a;2 _ 7 
X^ = 2x —3 
X3 = + 1. 
Hence, for x z= co, we have + + + + 
x= -ao '' _-f-__4. 

therefore, the equation has three real roots, two positive and one 
negative. 

To determine the initial figures of these roots, we have 

for a; = ... H h for re = . . . H h 

x = l ... + + xz=-l ... -] -f 

a; = 2...-|- + + + a;=- 2. ..+ + - + 

ar-^-3...+-|-- + 

hence there are two roots between 1 and 2, and one between 
— 3 and — 4. 

In order to ascertain the first figures 
in the decimal parts of the two roots 
situated between I and 2, we shall trans- 
form the preceding functions into others, 
in which the value of x is diminished by 
1. Thus, for the fimctiou X, we have 
this operation : 



1+0-7 + 7(1 
1 + 1-6 

1 -6+ 1 

1 +2 



And transforming the others in 
*he same way, we obtain the 
functions 



2-4 
1 

3 

r; = 3y2 + 6y -4; 
r, = 2i/ - 1 ; 
F3 rr + 1. 



CHAP. XIJ 


HORN 


ER'S METHC 


)D. dyo 


Let y = .1 vi 


have 


+ -- + 


two variations^ 


2, = .2 , 


C( 


+ + 


(( 


y = .3 


(( 


+ -- + 


(( 


y = .4 


(t 


1- 


one variation, 


2/ = .5 


(( 


=F -f 


u 


, 2/ = .6 


(C 


- + + + 


« 


2/ = .7 


(i 


+ + + + 


no variation. 


'llierefore the initial figures 


5 of the two 


30sitive roots are 1.3, 1.6, 


I et us now find 


the decimal part of the first root. 


1 hO 


-7 




4- 7 (1.356695867 


1 


1 




-6 


1 


-6 




*1 


1 


2 




-.903 


2 


*_4 


*^.097 


1 




99 


- .086625 



*3.3 
3 


3.6 
3 


**3.9 5 
5 


4.0 
5 


***4.0 56 
6 


4.0 62 
6 


•^►**4.0|68 8 
8 



|4.0|696 




-1.7325 

.2000 
***- 1.53 25 

.024336 
-1.508164 

.024372 
***- 1.4837912 

.0032514 

-1.4805318 
.0032514 



1 .4772 
.0003 



-1.4769 
,0003 



*«* .010375 

— .009048984 

**^* .001326016 
-.001184430 

.000141586 

— .000132923 

.000008663 

— .000007382 



.000001281 

- .000001181 
.000000100 

- .000000089 
.000000011 

- .000000010 



■-11 4|4|716|5 



396 ELEMENTS OF ALGEBRA. LCHAP. XI 

The operations in the example are performed as follows : 

1st. We find the places and the initial figures of the posi- 
tive roots, to include the first decimal place by Sturms' theorem, 

2d. Then to find the decimal part of the first positive root, 
we arrange the co-efficients, and perform a succession of trans 
formations by Synthetical Division, which must begin with the 
initial figures already known. 

We first transform the given equation into another whose 
roots shall be less by 1. The co-efficients of this new equa4;ion 
are, 1, 3, — 4 and 1, and are all, except the first, marked by 
a star. The root of this transformed equation, corresponding 
to the root sought of the given equation, is a decimal frac- 
tion of which we know the first figure 3. 

We next transform the last equation into another whose 
roots are less by three-tenths, and the co-efficients of the new 
equation are each marked by two stars. 

The process here changes, and we find the next figure of 
the root by dividing the absolute term .097 by the penulti- 
mate CO- efficient — 1.93, giving .05 for the next figure of the root. 

We again transform the equation into another whose roots 
shall be less by .05, and the co- efficients of the new equation 
are marked by three stars. 

We then divide the absolute term, .010375 by the penultimate 
co-efficient, — 1.5325, and obtain .006, the next figure of the 
root : and so on for other figures. 

In regard to the contractions, we may observe that, having 
decided on the number of decimal places to which the figures 
in the r* ;)t are to be carried, we need not take notice of 
figures which fall to the right of that number in any of the 
dividends. In the example under consideration, we propose to 
carry the operations to the 9th decimal place of the root ; 
hence, we may reject all the decimal places of the dividends 
after the 9th. 

The fourth dividend, marked by four stars, contains nine 
decimal places, and the next dividend is to contain no more. 



CHAP. XL] . HOKNEK's METHOD. 897 

But the corresponding quotient figure 8, is the fourth figure 
from the decimal point ; hence, at this stage of the operation, all 
the places of the divisor, after the 5th, may be omitted, since 
the 5th, multiplied by the 4th, will give the 9th order of deci- 
mals. Again : since each new figure of the root is removed 
one place to the right, one additional figjire, in each subsequent 
divisor, may be omitted. The contractions, therefore, begin by 
striking oflT the 2 in the 4th divisor. 

in passing from the first column to the second, in the next 
operation, we multiply by .0008 ; but since the product is to 
be limited to five decimal places, we need take notice of but 
one decimal place in the first column ; that is, in the first 
operation of contraction, we strike oflf, in the first column, the 
two figures 68 : and, generally, for each figure omitted in the 
second column^ we omit two in the first. 

It should be observed, that when places are omitted in either 
column, whatever would have been carried to the last figure 
retained, had no figures been omitted, is always to be added 
to that figure. Having found the figure 8 of the root, we need 
not annex it in the first column, nor need we annex any sub- 
sequent figures of the root, since they would all fall at the 
right, among the rejected figures. Hence, neither 8, nor any 
subsequent figures of the root, will change the available part 
of the first column. 

In the next operation, we divide .000141586 by 1.4772, omit- 
ting the figure 8 of the divisor: this gives the figure 9 of the 
root. We then strike oflf the figures 4.0, in the first column, 
and multiplying by .00009, we form the next divisor in the 
second column, — 1.4769, and the next dividend in the od 
cokmn, .000008663. Striking oflf 5 in this divisor, we find 
the next figure of the root, which is 5. 

It is now evident that the products from the first column, 
will fall in the second, among the rejected figures at the right ; 
we need, therefore, in future, take no notice of them. 

Omitting the right hand figure, the next divisor will be 1.476, 
and the next figure of the root 8. Then omitting 6 in the 



ELEMENTS OF ALGEBRA. 



LCHAP. XI. 



divisor, we obtain the quotient figure 8 : omitting 7 we obtain 
6, and omitting 4 -we obtain 7, the last fig are to be found. We 
have thus found the root x = 1.356895876 . . . . ; and all similajf 
examples are wrought after the same manner. 

Tlie next operation is to find the root whose initial figures are 
1.6, to nine decimal places. The operations are entirely slaiilar 
to those just explained. 

We find for the second root, x = 1.6920214'!. 

For the negative root, change the signs of the second and 
fourth terms (Art. 280), and we have. 






- 7 




- 7 (3.0489173396 


3 


9 




+ 6 


3 


2 




-1 


3 


18 
20 




.814464 


6. 


- .185536 


3 


, 


3616 


.166382592 



9.0 4 


4 


9.0 8 


4 


9.128 


8 


9.136 


8 



20.3616 
.3632 



20 



7248 
73024 



20.797824 
73088 



19153408 
18791228 

-362180 

.208875 

- 153305 
146212 



20.87091 

823 



.|9.1|44 



20.87914 
823 



20.8873 



20.8874i6 



2|0.!8|8|7|5 
Find the roots of the equation 

x^+ lla;2 - 102a; + 181 = 0. 



-7093 
6266 

-827 
626 

-201 

188 

12 



CHAP. XI.] HORNER'S METHOD. 399 

The functions are 

X = x^ + Wx' - \02x + 181 
Xi = 3^2 4. 22x - 102 
X2 = 122a; - 393 

-^3 = + ; 

and the signs of the leading terms are all + ; hence, the sub. 
stitution of — 00 and + ^ must give three real roots. 

To discover the situation of the roots, we make the substitu- 
tions • 

a; = which gives -\ 1- two variations 

x = \ " H h " 

x = 2 " . H h " 

x = ^ " H h « 

X = 4: " + + + +no variation ; 

hence the two positive roots are between 3 and 4, and we must 
therefore transform the several functions into others, in which £ 
shall be diminished by 3. Thus we have (Art. 314), 
F = y^ + 20y2 - % + 1 
Y, = 3y2 + 40y - 9 
Fs = 122y - 27 

Make the following substitutions in these functions, viz. : 



2/= 


dgns 


+ -- + 


two vauiations 


y = .i 


ii 


+ -- + 


u 


y = .2 


U 


+ — + 


(C 


y = .3 


u 


+ + + + 


no variation ; 



hence, the two positive roots are between 3.2 and 3.3, and wf 
must again transform the last functions into others, in which y 
shall be diminished by .2. Effecting this transformation, we have 

Z = z^ + 20.622 _ 882 _{_ ,008 

Z, = 3^2 -\- 41.22 - .88 

Z2 = 1222 - 2.6 

Z, = H-. 



400 ELEMKNTS OF ALGEBRA. [CHAP. XL 

Let z = then signs are H f- two variations, 

2 = .01 . " " H + 

z = .02 " " h one variation, 

s = M " " + + + + no variation ; 

nence we have 3.21 and 3.22 for the positive roots, and the sum 
of the roots is — 11 ; therefore, — 11 — 3.21 — 3.22 = — 17.4, 
is the negative root, nearly. 

For the positive root, whose initial figures are 3.21, we have 
X = 3.21312775 ; 
and for the root whose initial figures are 3.22, we have 

X = 3.229522121 ; 
and for the negative root, 

^ x= - 17.44264896. 

EXAMPLES. 

1. Find a root of the equation x^ -\- x"^ -\- x — 100 = 0. 

Ans. 4.2044299731. 

2. Find the roots of the equation x^ — 12^2 + 12.r — 3 == 0. 

-h 2.858083308163 



Ans. 



+ .606018306917 
-f .443276939605 
- 3.907378554685. 



3. Find the roots of the equation x^ — 82;^ + 14.t^ + 4a: — 8=^0 

r + 5.2360679775 



/\ti9 


\ + .7639320225 


1 +2.7320.508075 


. - .7320508075. 


1 Find the roots of the equation 


x^-\Ox^ + Qx+\=0. 




' - 3.0653157912983 




- .6915762804900 


Ans. ' 


- .1756747992883 




-f .8795087084144 


'4> 


, -f 3.0530581626622. 



D 559 



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